/ 


i^\t'0 


IN  MEMORIAM 
FLORIAN  CAJORl 


Vla^i 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/fiveplacelogaritOOwentrich 


MATHEMATICAL   TEXT-BOOKS 

By  G.  A.  WENTWORTH,  A.M. 

Mental  Arithmetic. 

Elementary  Arithmetic. 

Practical  Arithmetic. 

Primary  Arithmetic. 

Grammar  School  Arithmetic. 

High  School  Arithmetic. 

High  School  Arithmetic  (Abridged). 

First  Steps  in  Algebra. 

School  Algebra. 

College  Algebra. 

Elements  of  Algebra. 

Complete  Algebra. 

Shorter  Course  in  Algebra. 

Higher  Algebra. 

New  Plane  Geometry. 

New  Plane  and  Solid  Geometry. 

Syllabus  of  Geometry. 

Geometrical  Exercises. 

Plane  and  Solid  Geometry  and  Plane  Trigonometry. 

New  Plane  Trigonometry. 

New  Plane  Trigonometry,  with  Tables. 

New  Plane  and  Spherical  Trigonometry. 

New  Plane  and  Spherical  Trig.,  with  Tables. 

New  Plane  and  Spherical  Trig.,  Surv.,  and  Nav, 

New  Plane  Trig,  and  Surv.,  with  Tables. 

New  Plane  and  Spherical  Trig.,  Surv.,  with  Tables. 

Analytic  Geometry. 


PLANE    AND    SPHERICAL 


TEIGONOMETOY  AND  TABLES 


BY 

G.  A.  WENTWOETH,  A.M. 

AUTHOR   OF   A   SERIES    OF   TEXT -BOOKS    IN   MATHEMATICS 


REVISED     EDITION 


Boston,  U.S.A.,  and  London 

GINN    &    COMPANY,    PUBLISHEES 

1897 


Entered,  according  to  Act  of  Congress,  in  tM  year  1882,  by 

G.  A.  WENTWORTH 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


Copyright,  1895,  by  G.  A.  Wentwobth. 


(QJ\^5  ( 


PEEFACE. 


"TN  preparing  this  work  the  aim  has  been  to  furnish  just  so  much 
of  Trigonometry  as  is  actually  taught  in  our  best  schools  and 
colleges.  Consequently,  all  investigations  that  are  important  only 
for  the  special  student  have  been  omitted,  except  the  development  of 
functions  in  series.  The  principles  have  been  unfolded  with  the 
utmost  brevity  consistent  with  simplicity  and  clearness,  and  inter- 
esting problems  have  been  selected  with  a  view  to  awaken  a  real 
love  for  the  study.  Much  time  and  labor  have  been  spent  in  devising 
the  simplest  proofs  for  the  propositions,  and  in  exhibiting  the  best 
methods  of  arranging  the  logarithmic  work. 

The  author  is  under  particular  obligation  for  assistance  to  G.  A. 
Hill,  A.M.,  of  Cambridge,  Mass.,  to  Prof.  James  L.  Patterson,  of 
Schenectady,  N.Y.,  to  Dr.  F.  N".  Cole,  of  Ann  Arbor,  Mich.,  and  to 
Prof.  S.  F.  Norris,  of  Baltimore,  Md. 

G.  A.  WENTWORTH. 
Exeter,  N.H.,  July,  1895. 


iv]S05100 


COJ^TEISTTS. 


PLANE    TRIGONOMETRY. 

CHAPTER   I.     Functions  op  Acute  Angles  : 

Angular  measure,  page  1 ;  trigonometric  functions,  3 ;  representation 
of  functions  by  lines,  7  ;  changes  in  the  functions  as  the  angle  changes, 
10 ;  functions  of  complementary  angles,  11 ;  relations  of  the  functions 
of  an  angle,  12  ;  formulas  for  finding  all  the  other  functions  of  an 
angle,  when  one  function  of  the  angle  is  given,  15 ;  functions  of  45°, 
30°,  60°,  17. 

CHAPTER   11.     The  Right  Triangle: 

Given  parts  of  a  triangle,  19.  Solutions  without  logarithms,  19 
Case  I.,  when  an  acute  angle  and  the  hypotenuse  are  given,  19 
Case  II.,  when  an  acute  angle  and  the  opposite  leg  are  given,  20 
Case  III.,  when  an  acute  angle  and  an  adjacent  leg  are  given,  20 
Case  IV.,  when  the  hypotenuse  and  a  leg  are  given,  21 
Case  v.,  when  the  two  legs  are  given,  21.  General  method  of 
solving  a  right  triangle,  22  ;  solutions  by  logarithms ,  24  ;  area  of  the 
right  triangle,  26 ;  the  isosceles  triangle,  31 ;  the  regular  polygon,  33. 

CHAPTER   III.     Goniometry: 

Definition  of  goniometry,  36  ;  angles  of  any  magnitude,  36 ;  general 
definitions  of  the  functions  of  angles,  37  ;  algebraic  signs  of  the  func- 
tions, 39 ;  functions  of  a  variable  angle,  40  ;  functions  of  angles  greater 
than  360°,  42  ;  formulas  for  acute  angles  extended  to  all  angles,  43 ; 
reduction  of  the  functions  of  all  angles  to  the  functions  of  angles  in  the 
first  quadrant,  46  ;  functions  of  angles  that  differ  by  90°,  48 ;  functions 
of  a  negative  angle,  49  ;  functions  of  the  sum  of  two  angles,  51 ;  func- 
tions of  the  difference  of  two  angles,  53  ;  functions  of  twice  an  angle,  55; 
functions  of  half  an  angle,  55  ;  sums  and  differences  of  functions,  56. 

CHAPTER   IV.     The  Oblique  Triangle  : 

Law  of  sines,  60  ;  law  of  cosines,  62  ;  law  of  tangents,  62.  Solu- 
tions:  Case  I.,  when  one  side  and  two  angles  are  given,  64  ;   Case  II., 


VI  TRIGONOMETRY. 

when  two  sides  and  the  angle  opposite  to  one  of  them  are  given,  66 ; 
Case  III.,  when  two  sides  and  the  included  angle  are  given,  71 ;  Case 
IV.,  when  the  three  sides  are  given,  74 ;  area  of  a  triangle,  78. 

CHAPTER  V.     Miscellaneous  Examples  : 

Plane  Trigonometry,  82 ;   Goniometry,  99. 
Examination  Papers,  106. 

CHAPTER   VI.     Construction  of  Tables  : 

Logarithms,  117;  exponential  and  logarithmic  series,  120;  trigo- 
nometric functions  of  small  angles,  125 ;  Simpson's  method  of  con- 
structing a  trigonometric  table,  127 ;  De  Moivi-e's  theorem,  128 ; 
expansion  of  sinx,  cos(c,  and  tanx,  in  infinite  series,  132. 


SPHERICAL   TRIGONOMETRY. 
CHAPTER   VII.     The  Right  Spherical  Triangle  : 

Introduction,  135 ;  formulas  relating  to  right  spherical  triangles, 
137  ;  Napier's  rules,  141.  Solutions:  Case  I.,  when  the  two  legs  are 
given,  142  ;  Case  II.,  when  the  hypotenuse  and  a  leg  are  given,  142 ; 
Case  III.,  when  a  leg  and  the  opposite  angle  are  given,  143  ;  Case  IV., 
when  a  leg  and  an  adjacent  angle  are  given,  143  ;  Case  V.,  when  the 
hypotenuse  and  an  oblique  angle  are  given,  144;  Case  VI.,  when  the 
two  oblique  angles  are  given,  144.  The  isosceles  spherical  triangle,  149. 

CHAPTER   VIII.     The  Oblique  Spherical  Triangle  : 

Fundamental  formulas,  150 ;  formulas  for  half  angles  and  sides, 
152  ;  Gauss's  equations  and  Napier's  analogies,  154.  Solutions  :  Case 
I.,  when  two  sides  and  the  included  angle  are  given,  156;  Case  II., 
when  two  angles  and  the  included  side  are  given,  158  ;  Case  III.,  when 
two  sides  and  an  angle  opposite  to  one  of  them  are  given,  160 ;  Case 
IV. ,  when  two  angles  and  a  side  opposite  to  one  of  them  are  given, 
162;  Case  V.,  when  the  three  sides  are  given,  163;  Case  VI.,  when 
the  three  angles  are  given,  164.     Area  of  a  spherical  triangle,  166. 

CHAPTER   IX.     Applications  of  Spherical  Trigonometry: 

To  reduce  an  angle  measured  in  space  to  the  horizon,  170;  to  find 
the  distance  between  two  places  on  the  earth's  surface,  when  the 
latitudes  of  the  places  and  the  difference  in  their  longitudes  are  known, 
171 ;  the  celestial  sphere,  171 ;  spherical  co-ordinates,  174 ;  the  astro- 
nomical triangle,  176  ;  astronomical  problems,  177. 


PLAICE   TEIGONOMETEY. 


CHAPTER   I. 

TRIGONOMETRIC   FUNCTIONS   OP   ACUTE 
ANGLES. 

§  1.     Angular  Measure. 

As  lengths  are  measured  in  terms  of  various  conventional 
units,  as  the  foot,  meter,  etc.,  so  different  units  for  measuring 
angles  are  employed,  or  have  been  proposed. 

In  the  common  or  sexagesimal  system  the  circumference  of 
a  circle  is  divided  into  360  equal  parts.  The  angle  at  the 
centre  subtended  by  each  of  these  parts  is  taken  as  the  unit 
angle  and  is  called  a  degree.  The  degree  is  subdivided  into 
60  minutes,  and  the  minute  into  60  seconds.  A  right  angle  is 
equal  to  90  degrees. 

Note.  The  sexagesimal  system  was  invented  by  the  early  Babylonian 
astronomers  in  conformity  with  their  year  of  360  days. 

In  the  circular  system  an  arc  of  a  circle  is  laid  off  equal  in 
length  to  the  radius.  The  angle  at  the  centre  subtended  by 
this  arc  is  taken  as  the  unit  angle  and  is  called  a  radian. 

The  number  of  radians  in  360°  is  equal  to  the  number 
of  times  the  length  of  the  radius  is  contained  in  the  cir- 
cumference. It  is  proved  in  Geometry  that  this  number  is 
2 7r(7r  =  3.1416)  for  all  circles;  therefore  the  radian  is  the 
same  angle  in  all  circles. 


2  TRIGONOMETRY. 

Since  the  circumference  of  a  circle  is  2  tt  times  the  radius, 

27r  radians  =  360°,  and  tt  radians  =  180° ; 

180° 
therefore,         1  radian  = =  57°  17'  45" 

TT 

IT 

and  1  degree  =  -r-^  radian  =  0.017453  radian. 

By  the  last  two  equations  the  measure  of  an  angle  can  be 

changed  from  radians  to  degrees  or  from  degrees  to  radians. 

180° 
Thus,   2  radians  =  2  X  -^—  =  2  X  (57°  17'  45")  =  114°  35'  30". 

'  TT 

Note.  The  circular  system  came  into  use  early  in  tlie  last  century. 
It  is  found  more  convenient  in  the  higher  mathematics,  where  the  radians 
are  expressed  simply  as  numbers.  Thus  the  angle  tt  means  tt  radians, 
and  the  angle  3  means  3  radians. 

On  the  introduction  of  the  metric  system  of  weights  and  measures  at 
the  close  of  the  last  century,  it  was  proposed  to  divide  the  right  angle  into 
100  equal  parts  called  grades^  which  were  to  be  taken  as  units.  The 
grade  was  subdivided  into  100  minutes  and  the  minute  into  100  seconds. 
This  French  or  centesimal  system,  however,  never  came  into  actual  use. 


Exercise   I. 
[Assume  ;r  =  3.1416.] 

1.  Keduce  the  following  angles  to  circular  measure,  express- 
ing the  results  as  fractions  of  tt..  60°,  45°,  150°,  195°,  11°  15', 
123°  45',  37°  30'. 

2  3 

2.  How  many  degrees  are  there  in  -  ir  radians  ?  t  tt  radians  ? 

-  TT  radians  ?  77;  tt  radians  ?  3-r  tt  radians  ? 
o  lb  lo 

3.  What  decimal  part  of  a  radian  is  1°  ?  1'? 

4.  How  many  seconds  in  a  radian  ? 


^"" 


TRIGONOMETRIC    FUNCTIONS.  3 

5.  Express  in  radians  one  of  the  interior  angles  of  a  regular 
octagon  ;  dodecagon. 

6.  On  a  circle  of  50  ft.  radius  an  arc  of  10  ft.  is  laid  off ; 
how  many  degrees  does  the  arc  subtend  at  the  centre  ? 

7.  The  earth^s  equatorial  radius  is  approximately  3963 
miles.  If  two  points  on  the  equator  are  1000  miles  apart, 
what  is  their  difference  in  longitude  ? 

8.  If  the  difference  in  longitude  of  two  points  on  the  equator 
is  1°,  what  is  the  distance  between  them  in  miles  ? 

9.  What  is  the  radius  of  a  circle,  if  an  arc  of  1  foot  sub- 
tends an  angle  of  1°  at  the  centre  ? 

10.  In  how  many  hours  is  a  point  on  the  equator  carried 
by  the  earth's  rotation  on  its  axis  through  a  distance  equal 
to  the  earth's  radius  ? 

11.  The  minute  hand  of  a  clock  is  3|-  ft.  long  ;  how  far 
does  its  extremity  move  in  25  minutes?     [Take  7r  =  ^.] 

12.  A  wheel  makes  15  revolutions  a  second ;  how  long 
does  it  take  to  turn  through  4  radians  ?     [Take  ir  =  ^-.'] 

§  2.     The  Trigonometric  Functions. 

The  sides  and  angles  of  a  plane  triangle  are  so  related  that 
any  three  given  parts,  provided  at  least  one  of  them  is  a  side, 
determine  the  shape  and  the  size  of  the  triangle. 

Geometry  shows  how,  from  three  such  parts,  to  construct 
the  triangle  and  find  the  values  of  the  unknown  parts. 

Trigonometry  shows  how  to  compute  the  unknown  parts  of 
a  triangle  from  the  numerical  values  of  the  given  parts. 

Geometry  shows  in  a  general  way  that  the  sides  and 
angles  of  a  triangle  are  mutually  dependent.  Trigonometry 
begins  by  showing  the  exact  nature  of  this  dependence  in 
the  riffht  triangle,  and  for  this  purpose  employs  the  ratios 
of  its  sides. 


TRIGONOMETRY. 


Let  MAN  (Fig.  1)  be  an  acute  angle.     If  from  any  points 

B,  D,  F, in  one  of  its  sides 

perpendiculars  BC,  DE,  FG, 

are  let  fall  to  the  other  side,  then 
the  right  triangles  ABC,  ADE, 

AFG, thus  formed  have  the 

angle  A  common,  and  are  there- 
fore mutually  equiangular  and 
similar.  Hence,  the  ratios  of 
their  corresponding  sides,  pair  by 


C 

Fig.  1 
pair,  are  equal. 


That  is, 


AC^AE^AG      AC  _AE  _AG 
AB~  AD~  AF'     BC~  DE~  FG' 

These  ratios,  therefore,  remain  unchanged  so  long  as  the  angle 
A  remains  unchanged. 

Hence,  for  every  value  of  an  acute  angle  A  there  are  certain 
numbers  that  express  the  values  of  the  ratios  of  the  sides  in 
all  right  triangles  that  have  this  acute  angle  A. 

There  are  all  together  six  different  ratios : 

I.  The  ratio  of  the  opposite  leg  to  the  hypotenuse  is  called 
the  Sine  of  A,  and  is  written  sin  A. 

II.  The  ratio  of  the  adjacent  leg  to  the  hypotenuse  is  called 
the  Cosine  of  A,  and  written  cos  A. 

III.  The  ratio  of  the  opposite  leg  to  the  adjacent  leg  is 
called  the  Tangent  of  A,  and  written  tan  A. 

IV.  The  ratio  of  the  adjacent  leg  to  the  opposite  leg  is 
called  the  Cotangent  of  A,  and  written  cot  A. 

V.  The  ratio  of  the  hypotenuse  to  the  adjacent  leg  is  called 
the  Secant  of  A,  and  written  sec  A. 

VI.  The  ratio  of  the  hypotenuse  to  the  opposite  leg  is  called 
the  Cosecant  of  A,  and  written  esc  A. 

These  six  ratios  are  called  the  Trigonometric  Functions  of 
the  angle  A. 


/ 


TRIGONOMETRIC    FUNCTIONS.  O 

To  these  six  ratios  are  often  added  the  two  following  func- 
tions, which  also  depend  only  on  the  angle  A : 

VII.  The  versed  sine  of  ^  is  1  —  cos  A  and  is  written  vers  A. 

VIII.  The  cover sed  sine  oi  A  is  1  —  sin  A  and  is  written 
covers  A. 


In  the  right  triangle  ABC  (Fig. 
2)  let  a,  b,  c  denote  the  lengths  of 
the  sides  opposite  to  the  acute  an- 
gles A,  B,  and  the  right  angle  C, 
respectively,  these  lengths  being  all 
expressed  in  terms  of  a  common 
unit.     Then, 


a     opposite  leg 

sm  J  =  -=r 7 • 

c     hypotenuse 


tan  J  = 


a     opposite  leg 
b     adjacent  leg' 


cos^  = 


cot^  = 


b  adjacent  leg 

c  hypotenuse ' 

b  adjacent  leg 

a  opposite  leg' 


c      hypotenuse 
b     adjacent  leg' 


c      hypotenuse 

esc  -4 vT      1         ■ 

a     opposite  leg 


h      c-b 

vers  A=^L = . 

G  c 


a     c  — 

covers  ^=1 = 

c         c 


Exercise  IT. 

1.  What  are  the  functions  of  the  other  acute  angle  B  of 
the  triangle  ABC  (Fig.  2)  ? 

2.  If  ^  +  ^  =  90°,  prove 


sin  A  =  cos  Bf 
cos  A  =  sin  B, 
tan  A  =  cot  B, 
cot  A  =ta,nB, 


sec  A  =  CSC  B, 
esc  A  =  sec  B, 
vers  A  =  covers  B, 
covers  A  =  vers  B. 


6  TRIGONOMETRY. 

3.  Find  the  values  of  the  functions  of  A,  if  a,  b,  c  respec- 
tively have  the  following  values  : 

(i.)   3,    4,     5.      (iii.)  S,  15,  17.      (v.)  3.9,       8,         8.9. 
(ii.)  5,  12,  13.      (iv.)   9,  40,  41.     (vi.)  1.19,     1.20,    1.69. 

4.  What  condition  must  be  fulfilled  by  the  lengths  of  the 
three  lines  a,  b,  c  (Fig.  2)  in  order  to  make  them  the  sides  of 
a  right  triangle  ?     Is  this  condition  fulfilled  in  Example  3  ? 

5.  Find  the  values  of  the  functions  of  A,  if  a,  b,  c  respec- 
tively have  the  following  values  : 

(i.)  2mn,  m^  —  n^,  m^-\-n^,  (iii.)  pqr,  qrs,  rsp. 

...  ^     2x7/        ,       x^-\-ip'  ..    .    mn   mv    nr 

(ii.)  — ^,x-\-y, — ^^-  (iv.)  — , — , — 

^    ^   X  —  y  X  —  7/  ^     ^   pq     sq    ps 

6.  Prove  that  the  values  of  a,  b,  c,  in  (i.)  and  (ii.),  Example 
5,  satisfy  the  condition  necessary  to  make  them  the  sides  of 
a  right  triangle. 

7.  What  equations  of  condition  must  be  satisfied  by  the 
values  of  a,  b,  c,  in  (iii.)  and  (iv.),  Example  5,  in  order  that 
the  values  may  represent  the  sides  of  a  right  triangle  ? 

Compute  the  functions  of  A  and  B  when, 


8.  a  =  24,  &  =  143.  11.    a=4^^,    b=-J 2pq. 

9.  a  =  0.264,  c  =  0.26^.        12.    a=V^+^,    c=p^q. 
10.  &  =  9.5,  c  =  19.3.  13.   b=2^^,  c=p-\-q. 

Compute  the  functions  of  A  when, 

14.  a  =  2b,  16.    a-\-b  =  ^c. 

15.  a  =  fc.  17.    a  —  b  =  ^. 

18.  Find  a  if  sin  ^  =  f  and  c  =  20.5. 

19.  Find  b  if  cos  A  =  0.44  and  c  =  3.5. 

20.  Find  a  if  tan  ^  =  -y-  and  b  =  2j\. 


TRIGONOMETRIC    FUNCTIONS.  1 

21.  Find  5  if  cot  ^  =  4  and  a  =  17. 

22.  Find  c  if  sec  ^  =  2  and  b  =  20. 

23.  Find  c  if  esc  A  =  6.45  and  a  =  35.6. 

Construct  a  right  triangle  :  given, 

24.  c  =  6,   tan^  =  |.  '26.   b  =  2,   sin^  =  0.6. 

25.  a  =  3.5,   cos^=:f  -27.    b  =  ^,   csc^  =  4. 

28.  In  a  right  triangle,  c  =  2.5  miles,  sin  ^  =  0.6,  cos^  = 
0.8  ;  compute  the  legs. 
-  29.  Construct  (with  a  protractor)  the  A  20°,  40°,  and  70°; 
determine  their  functions  by  measuring  the  necessary  lines, 
and  compare  the  values  obtained  in  this  way  with  the  more 
nearly  correct  values  given  in  the  following  table  : 


20° 

40° 

70° 

sin 

cos 

tan 

cot 

sec 

CSC 

0.342 
0.643 
0.940 

0.940 
0.766 
0.342 

0.364 
0.839 
2.747 

2.747 
1.192 
0.364 

1.064 
1.305 
2.924 

2.924 
1.556 
1.064 

30.  Find,  by  means  of  the  above  table,  the  legs  of  a  right 
triangle  if  ^  ==  20°,  c  =  l;  also  if  ^  ==  20°,  c  =  4. 

31.  In  a  right  triangle,  given  a  =  S  and  c  =  5;  find  the 
hypotenuse  of  a  similar  triangle  in  which  a  =  240,000  miles. 

32.  By  dividing  the  length  of  a  vertical  rod  by  the  length 
of  its  horizontal  shadow,  the  tangent  of  the  angle  of  elevation 
of  the  sun  at  the  time  of  observation  was  found  to  be  0.82. 
How  high  is  a  tower,  if  the  length  of  its  horizontal  shadow  at 
the  same  time  is  174.3  yards  ? 


§  3.   Representation  of  the  Functions  by  Lines. 

The  functions  of  an  angle,  being  ratios,  are  numbers ;  but 
we  may  represent  them  by  lines  if  we  first  choose  a  unit  of 
length,  and  then   construct   right   triangles,  such  that  the 


8 


TRIGONOMETRY. 


Fig.  3. 


denominators  of  the  ratios  shall  be  equal  to  this  unit.     The 
most  convenient  way  to  do  this  is  as  follows  : 

About  a  point  0  (Fig.  3)  as 
a  centre,  with  a  radius  equal  to 
one  unit  of  length,  describe  a 
circle  and  draw  two  diameters 
AA^  and  BB^  perpendicular  to 
each  other. 

The  circle  with  radius  equal 
to  1  is  called  a  unit  circle,  AA^ 
the  horizontal^  and  BB^  the 
vertical  diameter. 

Let  AOP  be  an  acute  angle, 
and  let  its  value  (in  degrees,  etc.)  be  denoted  by  x.  We  may 
regard  the  Z  a;  as  generated  by  a  radius  OP  that  revolves 
about  0  from  the  position  OA  to  the  position  shown  in  the 
figure ;  viewed  in  this  way,  OP  is  called  the  moving  radius. 

Draw  PM  J.  to  OA,  PN  ±  to  OB.  In  the  rt.  A  0PM  the 
hypotenuse  0P  =  1;  therefore,  sinx  =^  PM-j  cos  x^=OM. 

Since  P3I  is  equal  to  ON,  and  ON  is  the  projection  of  OP 
on  BB',  and  since  OM  is  the  projection  of  OP  on  AA',  there- 
fore, in  a  unit  circle, 

sinic  =  projection  of  moving  radius  on  vertical  diameter; 

cos  07  =  projection  of  moving  radius  on  horizontal  diameter. 

Through  A  and  B  draw  tangents  to  the  circle  meeting  OP, 

produced  in  T  and  S,  respectively;  then,  in  the  rt.  A  OAT, 

the  leg  0A  =  1,  and  in  the  rt.  A  OBS,  the  leg  0B  =  1',  while 

the  Z  OSB  =  Zx.     Therefore, 


tainx  =  AT] 
seca:  =  OT: 


cot  x  =  BS', 
CSC  x  =  OS; 


veTsx  =  AM'j 
covers  x  =  BN. 


These  eight  line  values  (as  they  may  be  termed)  of  the 
functions  are  all  expressed  in  terms  of  the  radius  of  the  circle 
as  a  unit ;  and  it  is  clear  that  as  the  angle  varies  in  value  the 


TRIGONOMETRIC    FUNCTIONS.  9 

line  values  of  the  functions  will  always  remain  equal  numer- 
ically to  the  ratio  values.  Hence,  in  studying  the  changes  in 
the  functions  as  the  angle  is  supposed  to  vary,  we  may  employ 
the  simpler  line  values  instead  of  the  ratio  values. 

Exercise  III.        U^ 

1.  Eepresent  by  lines  the  functions  of  a  larger  angle  than 
that  shown  in  Fig.  3. 

If  X  is  an  acute  angle,  show  that 

2.  since  is  less  than  tancc. 

3.  seca;  is  greater  than  tana;. 

4.  csccc  is  greater  than  cot  jr. 
Construct  the  angle  x  if 

5.  tanir  =  3.  7.    cosa?  =  ^.  9.    sin  a;  =  2  cos  ic. 

6.  cscx  =  2.  8.    sin  ic^  cos  a;.         10.    4  sin  a;  =  tan  a;. 

11.  Show  that  the  sine  of  an  angle  is  equal  to  one-half  the 
chord  of  twice  the  angle. 

12.  Find  X  if  sin  x  is  equal  to  one-half  the  side  of  a  regular 
inscribed  decagon. 

13.  Given  x  and  y,  x-\-y  being  less  than  90°;    construct 
the  value  of  sin  (x-\-y)  —  sin  x. 

14.  Given  x  and  y,  x-\-y  being  less  than  90°;  construct 
the  value  of  tan  (a?  +  y)  —  sin  (x-\-y)-\-  tana?  —  sina;. 

Given  an  angle  x ;  construct  an  angle  y  such  that 

15.  sin?/ =  2  sin  a?.  17.    tan?/ =  3  tan  a?. 
!  16.    cos  ?/ =:  J  cos  ar.                         18.    sec?/  =  csca;. 

19.  Show  by  construction  that  2  sin  ^  >  sin  2  A. 

20.  Given  two  angles  A  and^,  A-\-B  being  less  than  90°; 
show  that  sin  (A-\-  B)<,  sin  A  -\-  sin  B . 

21.  Given  sin  a;  in  a  unit  circle;  find  the  length  of  a  line 
corresponding  in  position  to  sin  a;  in  a  circle  whose  radius  is  r. 

22.  In  a  right  triangle,  given  the  hypotenuse  c,  and  also 
sin A  =  m,  cos  A  =  n\  find  the  legs. 


10 


TRIGONOMETRY. 


§  4.    Changes  in  the  Functions  as  the  Angle  Changes. 

If  we  suppose  the  /_  AOP,  or  x  (Fig.  4)  to  increase  gradu- 
ally by  the  revolution  of  the  moving  radius  OP  about  0, 
the  point  F  will  move  along  the  arc 
AB  towards  B,  T  will  move  along 
the  tangent  AT  away  from  A,  S  will 
move  along  the  tangent  BS  towards 
B,  and  M  will  move  along  the  radius 
OA  towards  0. 

Hence,  the  lines  FM,  AT,  OT  will 
gradually  increase  in  length,  and  the 
lines  OM,  BS,  OS  will  gradually 
decrease.     That  is, 

As   an   acute   angle   increases,   its 
sine,  tangent,  and  secant  also  increase, 
while  its  cosine,  cotangent,  and  cose- 
cant decrease. 
On  the  other  hand,  if  we  suppose  x  to  decrease  gradually, 
the  reverse  changes  in  its  functions  will  occur. 

If  we  suppose  x  to  decrease  to  0°,  OF  will  coincide  with  OA 
and  be  parallel  to  BS.     Therefore,  FM  and  AT  will  vanish, 
OM  will  become  equal  to  OA,  while  BS  and  OS  will  each  be 
infinitely  long,  and  be  represented  in  value  by  the  symbol  oo. 
And  if  we  suppose  x  to  increase  to  90°,  OF  will  coincide 
with  OB  and  be  parallel  to  AT.     Therefore,  PJf  and  OS  will 
each  be  equal  to  OB,  OM  and  BS  will  vanish,  while  AT  and 
OT  will  each  be  infinite  in  length.         ^ 
Hence,  as  the  angle  x  increases  from  0°  to  90°, 
sinx  increases  from  0  to  1, 
cos  x  decreases  from  1  to  0, 
tan  X  increases  from  0  to  oo, 
cotcc  decreases  from  go  to  0, 
sec  a;  increases  from  1  to  go, 
CSC  a;  decreases  from  oo  to  1. 


TRIGONOMETRIC    FUNCTIONS. 


11 


The  values  of  the  functions  of  0°  and  of  90°  are  the  limiting 
values  of  the  functions  of  an  acute  angle.  It  is  evident  that 
(disregarding  the  limiting  values), 

Sines  and  cosines  are  always  less  than  1; 

Secants  and  cosecants  are  always  greater  than  1 ; 

Tangents  and  cotangents  have  all  values  between  0  and  oo. 

Remark.  We  are  now  able  to  understand  why  the  sine,  cosine,  etc., 
of  an  angle  are  called  functions  of  the  angle.  By  a  function  of  any  mag- 
nitude is  meant'' another  magnitude  which  remains  the  same  so  long  as 
the  first  magnitude  remains  the  same,  but  changes  in  value  for  every 
change  in  the  value  of  the  first  magnitude.  This,  as  we  now  see,  is  the 
relation  in  which  the  sine,  cosine,  etc.,  of  an  angle  stand  to  the  angle. 


/- 


§  5.    Functions  of  Complementary  Angles. 


The  general  form  of  two  complementary  angles  is  A  and 
90° -A 

In  the  rt.  A  ABC  (Fig.  5), 
^  -f  ^  =  90°;  hence  ^  =  90°  —  A. 
Therefore  (§  2), 
sin  A  =  cos  B  =  cos  (90°  —  A), 
cos  ^  =  sin  ^  ==  sin  (90°  —  A), 
tan  ^  =  cot  i?  =:  cot  (90°  —  A), 
cot  ^  =  tan  ^  =  tan  (90°  —  A), 
sec  ^  =  CSC  ^  =  esc  (90°  —  A), 
CSC  A  =  sec  B  =  sec  (90°  —  A). 
Therefore, 

^ach  function  of  an  acute  angle  is  equal  to  the  co-named 
function  of  the  complementary  angle. 

Note.  Cosine,  cotangent,  and  cosecant  are  sometimes  called  co- 
functions;  the  words  are  simply  abbreviated  forms  of  complemenVs  sine, 
complemenVs  tangent,  and  complemenVs  secant. 

Hence,  also, 

Ang  function  of  an  angle  between  45°  and  90°  may  he  found 
hy  taking  the  co-named  function  of  the  complementary  angle 
between  0°  and  45°. 


12  TRIGONOMETRY. 


Exercise  IV. 


1.  Express  the  following  functions  as  functions  of  the 
complementary  angle: 

sin  30°.  tan  89°.  esc  18°  10'.  cot  82°  19'. 

cos  45°.  cot  15°.  cos  37°  24'.  esc  54°  46'. 

2.  Express  the  following  functions  as  functions  of  an 
angle  less  than  45°: 

sin  60°.  tan  57°.  esc  69°  2'.  cot  89°  59'. 

cos  75°.  cot  84°.  cos  85°  39'.  esc  45°  1'. 

3.  Given  tan  30°  =  -J  V3 ;  find  cot  60°. 

4.  Given  tan  ^  =  cot  ^  ;  find  A. 

5.  Given  cos  ^  =  sin2^;  find  A ^:, 

6.  Given  sin  ^  =  cos  2  ^ ;  find  A.    :■ 

7.  Given  cos  A  =  sin  (45°  —  ^A);  find  A. 

8.  Given  cot  ^  ^  =  tan  A ;  find  A.^ 

9.  Given  tan  (45°  -\-A)=GotA',  find  A. 

10.  Find^if  sin^  =  cos4A   ^ 

11.  Find  A  if  cot  ^  =  tan  8  ^.    : 

12.  Find  A  if  cot  A  =  tan  nA. 

§  6.     Eelations  of  the  Functions  of  an  Angle. 
Formula  [1].     Since  (Fig.  5)  a^  +  b^=c^,  therefore, 


7^+?  =  ^     ^^ 


(!)■-(')■ 


Therefore  (§  2),  (sin  Af  +  (cos  Ay  =  l'j 
or,  as  usually  written  for  convenience, 

sin2A  +  cos2A  =  l.  [1] 

That  is  :  The  sum  of  the  squares  of  the  sine  and  the  cosine  of 
an  angle  is  equal  to  unity. 


TRIGONOMETRIC    FUNCTIONS.  13 

Formula  [1]  enables  us  to  find  the  cosine  of  an  angle  when 
the  sine  is  known,  and  vice  versa.  The  values  of  sin  A  and 
of  cos  A  deduced  from  [1]  are  : 


sm 


^  =  V 1  —  cos^^,     cos  ^  =  V 1  —  sin^A 


Formula  [2],     Since 

a       h a      c a 

c  '   c       c      b      b 

therefore  (§  2),  tan  A  =  ^^-  [2] 

That  is  :  The  tangent  of  an  angle  is  equal  to  the  sine  divided 
by  the  cosine. 

Formula  [2]  enables  us  to  find  the  tangent  of  an  angle 
when  the  sine  and  the  cosine  are  known. 


Formula  [3].     Since 

-X-  =  l,    -X-  =  l,     and     -X-  =  l, 

c       a  c      b  a      a 

therefore  (§  2),  sin  A  X  esc  A  =  1  ] 

cosAXsecA=l>  C^] 

tan  AX  cotA=l  J 


That  is :  The  sine  and  the  cosecant  of  an  angle,  the  cosine 
and  the  secant,  and  the  tangent  and  the  cotangent,  pair  by 
pair,  are  reciprocals. 

The  equations  in  [3]  enable  us  to  find  an  unknown  func- 
tion contained  in  any  pair  of  these  reciprocals  when  the  other 
function  in  this  pair  is  known. 


zy\ 


14  TRIGONOMETRY. 

Exercise  V. 

1.  Prove  Formulas  [1]  -  [3],  using  for  the  functions  the 
line  values  in  the  unit  circle  given  in  §  3. 

Prove  that 

2.  l  +  tan2A  =  sec2A. 

3.  l  +  cot2A  =  csc2A. 

Note.      Equations  2  and  3  should  be  remembered. 

4.  cot  A  =  — -' 

sm^ 

5.  sin  ^  sec  ^  =  tan  A. 

6.  sin  A  cot  A  =  cos  A. 

7.  cos  ^  esc  ^  =  cot  A. 

8.  tan  ^  cos  ^  =  sin  A. 

9.  sin  A  sec  A  cot  ^  =  1. 

10.  cos  A  CSC  A  tan  J  =  1. 

11.  (l-sin2^)tan2J  =  sin2^. 

12.  Vl  —  cos^^  cot  A  =  cos  A. 

13.  (l  +  tan2^)sin2^  =  tan2A 

14.  csc2^(l  — sin2^)  =  cot2J. 

15.  tan2^cos2^+cos2^  =  l. 

16.  (sin^^  —  cos^Af  =  1  —  4  sin^^  cos»^ 
■  17.  (1  — tanM)2  =  sec*^--4tan2A 

^o     sin^  ,   cos^ 
lo-   - — 7  +  - — 7  =  sec  A  CSC  J. 
cos^      sm^ 

19.  sin^^  —  cos^^  =  sin^^  —  cos^^. 

20.  sec^  — cos^=sin^  tanA 

21.  esc  A  —  sin  A  =  cos  A  cot  A, 
cos  A         1  +  sin^ 


^/ 


^/22. 


1  — sin^  cos^ 


/ 


\'' 


TRIGONOMETRIC    FUNCTIONS.  15 


§  7.     Application  of  Formulas  [1]  -  [3]. 

Formulas  [1],  [2],  and  [3]  enable  us,  when  any  one  func- 
tion of  an  angle  is  given,  to  find  all  the  others.  A  given  value 
of  any  one  function,  therefore,  determines  all  the  others. 

Example  L    G-iven  sin  ^  =  | ;  find  the  other  functions. 


By  [IJ,  cos  ^  =  Vl  - 1  -  ^i=h-^^' 

By  [2],  tan^  =  |^|V5^?X^^^  =  |V5. 

By  [3],  cot  ^  =  -y-,  sec^=  -  V5,   csc^  =  -- 

Example  2.    Given  tan  A  =  3  -,  find  the  other  functions. 
By  [2],         ^  =  3. 

•^    •-    -"  COS  ^  . 

And  by  [1],  sinM  +  cos^^  =  1. 

If  we  solve  these  equations  (regarding  sin  A  and  cos  A  as 
two  unknown  quantities),  we  find  that, 

sin^  =  3V^,    cos^  =  V^- 
Then  by  [3],  cot  A  =  ^,   seGA  =  VlO,    csc  yl  =  ^  VlO. 

Example  3.    Given  sec  A  =  m;   find  the  other  functions. 


By  [3], 

COS  A  = 

sin^  = 

tan^  = 
CSC  A  = 

m 

_1 
m 
1 

=v>- 

1 

cot  4  '■ — 

-1_ 

By  [1], 

Vm2- 

-1. 

By  [2],  [3], 

m 

-1, 

Vm2- 

^' 

Vm^-l 


16  TRIGONOMETRY. 

Exercise  VI. 

Eind  the  values  of  the  other  functions,  when 

1.  sin  A  =  If.         5.    tan  A  =  ^.         9.    esc  A  =  V2. 

2.  sin  ^  =  0.8.        6.    cot^=::l.        10.    sin  ^  =  w. 

2  m 

3.  cos  A  =  If.      ^  7.    cot  A  =  0.5.    11.    sin  A  =  —j- — -  • 

4.  cos  ^  =  0.28.      8.    sec  ^==2.       12.    cos  ^  = -IT^* 

nr  -f-  n^ 

Given  tan  45°  =  1 ;  find  the  other  functions  of  45°. 
Given  sin  30°  =  i;  find  the  other  functions  of  30°. 
Given  esc  60°  =  §  V3.  find  the  other  functions  of  60°. 
'Vie.    Given  tan  15°=  2— V3;  find  the  other  functions  of  15°. 

17.  Given  cot  22°  30'=  V24-I;  find  the  other  functions 
of  22°  30'. 

18.  Given  sin     0°  =0;  find  the  other  functions  of  0°. 

19.  Given  sin  90°  =  1;  find  the  other  functions  of  90°. 
T  20.    Given  tan  90°  =  co  ;  find  the  other  functions  of  90°. 

21.  Express  the  values  of  all  the  other  functions  in  terms 
of  sin  A. 

22.  Express  the  values  of  all  the  other  functions  in  terms 
of  cos  A. 

23.  Express  the  values  of  all  the  other  functions  in  terms 
of  tan  A. 

24.  Express  the  values  of  all  the  other  functions  in  terms 
of  cot  A. 

25.  Given  2  sin  A  =  cos  A  ;  find  sin  A  and  cos  A. 
4.  26.  Given  4  sin  A  =  tan  A ;  find  sin  A  and  tan  A. 
-V  27.    If  sin  A :  cos  A  =  9  :  40,  find  sin  A  and  cos  A. 

■  28.    Transform  the  quantity  tanM  +  cot^^  —  sin^^l  —  cos^^ 
into  a  form  containing  only  cos  A. 
\29.    Prove  that  sin  A -{-cos  ^  =  (1  +  tan  A)  cos  A. 
30.    Prove  that  tan  ^  +  cot  ^  =  sec  ^  X  esc  A. 


TRIGONOMETRIC    FUNCTIONS. 


17 


§  8.     Functions  of  45°. 


Let  ABC  (Fig.  6)  be  an  isosceles  right  triangle,  in  whicli 
the  length  of  the  hypotenuse  AB  is 
equal  to  1;  then  AC  is  equal  to  BC^ 
and  the  angle  A  is  equal  to  45°. 
Since  AC^  +  BC^  =  1,  therefore 
2  AC^  =  1,  and  ^C  =  V|  =  i  V2. 
Therefore  (§  2), 

sin45°=:cos45°  =  iV2. 
tan45°  =  cot45°=:l._ 
sec  45°  =  esc  45°  =  V2. 


§  9.     Functions  of  30°  and  60' 


Let  ABC  (Fig.  7)  be  an  equilateral  triangle,  in  which  the 
length  of  each  side  is  equal  to  1 ;  and  let  CD  bisect  the  angle 
C.  Then  CD  is  perpendicular  to  AB  and  bisects  AB  ;  hence, 
AD  =  i,  and  Ci)=  Vl -i==  Vf  =  i  V3. 

In  the  right  triangle  ADC,  the  angle  ACD  =  SO°,  and  the 
angle  CAD  =  60°.         Whence  (§  2), 

sin  30°  =  cos  60°  =  -J. 
cos  30°  =  sin  60°  =  J  V3. 

tan  30°  =  cot  60°  =  -^^  =  i^/S. 

V3 

cot  30°  =  tan  60°  =V3. 

sec  30°  =  esc  60°  =  4=  =  f  V3. 

V3 
CSC  30°  =  sec  60°  =  2. 


The  results  for  sine  and  cosine  of  30°,  45°,  and  60°  may  be 
easily  remembered  by  arranging  them  in  the  following  form : 


18 


TRIGONOMETRY. 


Angle  .... 

30° 

45° 

60° 

^  Vi  =  0.5 

Sine 

iVi 

iV2 

^V3 

iV2  =  0.70711 

Cosine   .  .  . 

^V3 

^V2 

Wi 

^V3  =  0.86603 

Exercise  VII. 
Solve  the  following  equations  : 


1.  2  cos  X  =  sec  X. 

2.  4  sin  X  =  CSC  x. 

3.  tan x  =2  sin  x. 

4.  sec  X  =  V2  tan  x. 

5.  sin^  ic  =  3  cos^  x. 

6.  2  sin^a;  + cos^a;  =  |. 


7.  3  tan^a:  — sec2a;  =  l. 

8.  tan  x  +  cot  £c  =  2. 

9.  sin^  a:;  —  cos  x  =  ^. 

10.  tan^  ic  —  sec  x  =  l. 

11.  sin   a?  +  V3  cos  ic  =  2. 

12.  tan^  X  -{-  csc^  x  =  3. 


^\\ 


13.  2  cos  X  -\-  sec  cc  =  3. 

14.  cos^  X  —  sin^  x  =  sin  x. 

A 15.  2  sin  cc  -j-  cot  £c  =  1  +  2  cos  x. 

16.       sin^  X  -\-  tan^  a:  =  3  cos^  x. 
^  17.       tan  X  -\-2  cot  a;  =  |  esc  a;. 

Note.  Went  worth  &  Hill's  Five-place  trigonometric  and  logarithmic 
tables  have  full  explanations,  and  directions  for  using  them.  Before  pro- 
ceeding to  Chapter  II.  the  student  should  learn  how  to  use  these  tables. 

Table  VI.  is  to  be  used  in  solutions  without  logarithms.  This  four- 
place  table  contains  the  natural  functions  of  angles  at  intervals  of  Y. 
The  decimal  point  must  be  inserted  before  each  value  given,  except 
where  it  appears  in  the  values  of  the  table. 


CHAPTER   II. 

THE    RIGHT    TRIANGLE. 

§  10.     The  Given  Parts. 

In  order  to  solve  a  right  triangle,  two  parts  besides  the 
right  angle  must  be  given,  one  of  them  at  least  being  a  side. 
The  two  given  parts  may  be : 

1.  An  acute  angle  and  the  hypotenuse. 
II.    An  acute  angle  and  the  opposite  leg. 

III.  An  acute  angle  and  the  adjacent  leg. 

IV.  The  hypotenuse  and  a  leg. 
y.   The  two  legs. 

§  11.     Solution  without  Logarithms. 

The  following  examples  illustrate  the  process  of  solution 
when  logarithms  are  not  employed. 

Case  I. 
Given  ^  =  43°  17',  c  =  26  ;  find  B,  a,  h, 

\.   ^  =  90° -J  =  46° 43'. 

2.  -  =  sin^:  .•.«  =  <?  sin  ^. 
c 

3.  -  =  cos  ^  :.*.&  =  coos  J. 
c 


20 


1/ 

sin^=    0.6856 

ij 

XlU^Ul-N  I. 

cosA=    0.7280 

c=          26 

G=          26 

41136 

43680 

13712 

14560 

a  =  17.8256 

b  =  18.9280 

Case  II. 

Given  ^  =  13°  58',  a  = 

:  15.2  ;  find  B,  b,  c. 

y 

B 

1.    i?  =  90°-J  =  76°2'. 

C/^ 

a 

2.     -  =  cotA:  .\b  =  acotA 
a 

y<^ 

0 

3. 

a        .      ,                    a 

/^    1 

h 

c                                 am  j± 

Fig.  9. 

cot  A  =       4.0207 

a  =  15.2,  sin  ^  =  0.2414 

a=          15.2 

0.2414)15.200(62.9 

80414 

14  484 

201035 

7160 

40207 

4828 

5  =  61.1146^ 

i 

c  =  62.9             2332 

Case  III. 
Given  ^  =  27°  12',  b  =  ^l',  find  B,  a,  c. 


B 


1.  5  =  90°  — ^  =  62°  48'. 

2.  y  =  tan^;  .'.a  =  bt2in  A. 

3.  -  =  cos^;  .  .c  = 


b 
Fia.  10. 


cos^ 


THE    RIGHT    TRIANGLE. 


21 


tan^=    0.5139 

h=     31 

5139 
15417 
a  =  15.9309 


h  =  31,  cos  A  =  0.8894 
0.8894)31.000(34.9 
26  682 


34.9 


4  3180 

3  5576 

7604 


Case  IV. 
Given  a  =  47,  c  =  63  ;  find  A,  B,  b. 


1.  sin^ 


a 


2.  ^  =  90°  — A 

3.  h  =  ^c^  —  0^=^ {g  —  a)(G-{- a).  ^ 


ct  =  47,  c  =  63. 

63)47.0(0.7460 
441 
2  90 

2  52 


c-\-a=    110 

c  —  a^=      16 

"660 

110 

^2=1760 


sin  ^  =  0.7460       380 

.-.J  =  48°  15'       378 

J5  =  4r45'  2 


Case  V. 
Given  a  =  121,  b  =  S7  ',  find  A,  B,  c. 


b  =  V1760 
=  41.95 


1.    ta,nA  = 

2. 

3.  c=V^M^- 


b 
■B  =  90' 


A. 


22 


TBIGONOMETRY. 


a  =  121,b  =  S7. 

37)121(3.2703 
111 
100 
74 
tan  ^  =  3.2703     260 
.-.^  =  73°  259 

^  =  17°  1 


a2  =  14641 
Z,2=^_1369 
c2=  16010 


.c  =Vl6010 
=  126.5 


§  12.    General  Method  of  Solving  the  Eight  Triangle. 

From  these  five  cases  it  appears  that  the  general  method  of 
finding  an  unknown  part  in  a  right  triangle  is  as  follows  : 

Choose  from  the  equation  A-\-B  =  90°,  and  the  equations  that 
define  the  functions  of  the  angles,  an  equation  in  which  the 
required  part  only  is  unknown ;  solve  this  equation,  if  neces- 
sary, to  find  the  value  of  the  unknown  part ;  then  compute  the 
value. 

Note.  In  Case  IV.,  if  the  given  sides  (here  a  and  c)  are  nearly  alike  in 
value,  then  A  is  near  90°,  and  its  value  cannot  be  accurately  found  from 
the  tables,  because  the  sines  of  large  angles  differ  little  in  value  (as  is 
evident  from  Fig.  4).  In  this  case  it  is  better  to  find  B  first,  by  means  of 
a  formula  proved  later.     See  formula  [18],  §  30  ;  viz., 


tani^ 


Example.    Given  a  =  49,  c  =  50  ;  find  A,  B,  b. 


c  —  a=l,  c  +  a 

^^  =  0.01010 
c  +  a 

tan  iJ5  =  0.1005 

.•.i5=5°44' 

B  =  11°  28' 

A  =  78°  32' 


c  —  a  =  1 
c  +  g  ==  99 
62  =  99 
b  =V99 
=  9.95 


THE    RIGHT    TRIANGLE. 


23 


Exercise  VIII. 

1.  In  Case  II.  give  another  way  of  finding  c,  after  b  has 
been  found. 

2.  In  Case  III.  give  another  way  of  finding  c,  after  a  has 
been  found. 

3.  In  Case  IV.  give  another  way  of  finding  h,  after  the 
angles  have  been  found. 

4.  In  Case  V.  give  another  way  of  finding  c,  after  the 
angles  have  been  found. 

5.  Given  B  and  c  ;  find  A,  a,  h. 

6.  Given  B  and  h  ;  find  A,  a,  c. 

7.  Given  B  and  a ;   find  A,  b,  c. 

8.  Given  b  and  c  ;  find  A,  By  a. 

Solve  the  following  right  triangles: 


9 

Given. 

Required. 

a  =3, 

6=4. 

A  =  36°  52', 

I?  =53°  8', 

c  =  5. 

10 

a=7. 

c  =  13. 

^=32°  35', 

B  =  57°  25', 

6=10.954. 

11 

a  =5.3, 

^  =  12°  17'. 

B  =  77°  43', 

6  =  24.342, 

c  =  24.918. 

12 

a  =10.4, 

i?  =  43°18'. 

A  =  46°  42', 

6  =  9.800, 

c  =  14.290. 

13 

c=2G, 

^  =  37°  42'. 

J5  =  52°18', 

a  =  15.900, 

6  =  20.572. 

14 

c  =  140, 

I?  =24°  12'. 

A  =  65°  48', 

a  =  127.694 

6  =  57.386. 

15 

6=19, 

c  =  23. 

^  =  34°  18', 

^=55°  42', 

a  =12.961. 

16 

6=98, 

c=  135.2. 

A  =  43°  33', 

J5=46°27', 

a  =  93.139. 

17 

6  =  42.4, 

A  =  32°  14'. 

i?  =  57°  46', 

a  =  26.733, 

c=  50.124. 

18 

6  =  200, 

5  =46°  11'. 

A  =  43°  49', 

a  =191.900 

c  =  277.160. 

19 

a  =95, 

6  =  37. 

A  =  68°  43', 

i?=21°17'. 

c=  101.951. 

20 

a  =  Q, 

c  =  103. 

^  =  3°  21', 

^=86°  39', 

6  =  102.825. 

21 

a  =  3.12, 

1?=5°8'. 

A  =  84°  52', 

6  =  0.280, 

c  =  3.133. 

22 

a  =17, 

c=18. 

^  =  70°  48', 

B  =  19°  12', 

6=5.916. 

23 

c=  57, 

A  =  38°  29'. 

B  =  51°  31', 

a  =  35.471, 

6  =  44.620. 

24 

a+c=18. 

6=12. 

A  =  22°37', 

5=  67°  23', 

a=5,c=13. 

25 

a  +  6=9, 

c=8. 

^  =  82°  18', 

5=  7°  42', 

ffl=  7.928, 
6=1.072. 

24 


TRIGONOMETRY. 


§  13.    Solution  by  Logarithms. 
Case  I. 
Given  A  =  34°  28',  c  =  18.75  ;  find  B,  a,  h. 

1.    ^=90° -^  =  55° 32'. 


2.    -=sin^:  .'.tt^csin^. 
c 


3.    -  =cos^:  .\h=cco^A. 
a 


log  a        =  log  c  +  log  sin  A 
logc        =    1.27300 
log  sin  ^=    9.75276  —  10 
log  a        =    1.02576 
a        =  10.611 


log&         =  log  c  +  log  cos -4 
logc         =    1.27300 
log  cos  ^=    9.91617  —  10 
log^»        =    1.18917 
b        =  15.459 


Case  II. 
Given  A  =  62°  10',  a  =  78  ;  find  B,  b,  c. 
B 


1.  i?==90°-^  =  27°50'. 

2.  -  =GotA:  .'.b^acotA. 
a 

3.  -  =sinA 


a  =  c  sin  A,  and  c  = 


sin^ 


log  b        =  log  a  -\-  log  cot  A 
log  a        =    1.89209 
log  cot  ^=    9.72262-10 
log^»        =    1.61471 
b        =  41.182 


logc  =loga-|-cologsin^ 

log  a  =    1.89209 

cologsin^=   0.05340 
logc  =   1.94549 

c  =88.204 


THE    RIGHT    TRIANGLE. 


26 


Case  III. 
Given  A  =  50°  2',  b  =  SS;  find  ^,  a,  c. 

1.  ^  =  90°  — ^  =  39°  58'. 

2.  -=tan^;  .-.a  =  6  tan  A 

3.  -  =G0SJ. 

G 


■,  J)  =zcQ,o^A,  and  c  = 


cos^ 


log  a        =  log  Z*  +  log  tan  A 
logh         =      1.94448 
log  tan  J  =    10.07670  —  10 
log<^        =      2.02118 
a        =  105.0 


logC  =:log^  +  COlogCOS^ 

log^  =     1.94448 

cologcos^=     0.19223 

logc  =     2.13671 

c  =137.0 


Case  IV. 
Given  c  =■  58.40,  a  =  47.55  ;  find  A,  B,  h. 


1.  sinJ=-- 

c 

2.  ^  =  90°  — A 

3.  -=cot^:     .'.b  =  aGotA. 


log  sin  A  =  log  a  +  colog  c 
log  a        =1.67715 
colog  c     =8.23359  —  10 
log  sin  y1=:  9.91074  — 10 

^  =  54°  31' 

^  =  35°  29' 


log  b        =  log  a  +  log  cot  A 
log  a        =    1.67715 
log  cot  ^=    9.85300-10 
logb        =   1.53015 
b        =  33.896 


26 


TRIGONOMETRY. 


Case  V. 


T 


Given  a  =  40, 

/ 

27; 

B 

find  A,  B,  c. 
1.   tan^  =  ^- 

0 

-} 

a 
0 

2.  ^=-90°-J. 

3.  -  =  sin  A. 

c 

.*.  a  =csin^;    . 

6    ^^ 
Fig.  17. 

sin  ^ 

log  tan  A  =  log  a  +  colog  h 
log  a        =   1.60206 
colog  b    =   8.56864-10 
logtan^  =  10.17070  — 10 
^  =  55°  59' 

^=34°r 


logc  ^log<x+ colog  sin^ 

logd  =  1.60206 

colog  sin  J  =  0.08152 
logc  =  1.68358 

c  =48.259 


Note.  In  Cases  IV.  and  V.  the  unknown  side  may  also  be  found  from 
the  equations 

(for  Case  IV.)        6  =  V^aTT^i  =  V(c  +  a){c-  a) ; 
(for  Case  V.)  c  =  VoH^. 

These  equations  express  the  values  of  h  and  c  directly  in  terms  of  the 
two  given  sides ;  and  if  the  values  of  the  sides  are  simple  numbers  {e.g.  5, 
12, 13),  it  is  often  easier  to  find  6  or  c  in  this  way.  But  tliis  value  of  c  is 
not  adapted  to  logarithms,  and  this  value  of  h  is  not  so  readily  worked  out 
by  logarithms  as  the  value  of  6  given  under  Case  IV.     See  also  §  12,  Note. 

§  14.    Area  of  the  Right  Triangle. 

It  is  shown  in  Geometry  that  the  area  of  a  triangle  is  equal 
to  one-half  the  product  of  the  base  by  the  altitude. 

Therefore,  if  a  and  h  denote  the  legs  of  a  right  triangle, 
and  F  the  area,  ^  _  i  ^^ 

By  means  of  this  formula  the  area  may  always  be  found 
when  a  and  b  are  given  or  have  been  computed. 


THE    RIGHT    TRIANGLE. 


27 


For  example  :  Find  the  area,  having  given : 


Case  I.  (§  13). 
^  =  34° 28',  c  =  18.75. 
First  find  (as  in  §  13)  log  a 
and  log^. 

log  i^"  =  log  a  +  log  h  +  colog  2 
log  a     =   1.02576 
log^*      =   1.18917 
colog2==  9.69897-10 

logi^=   1.91390 
F         =82.016 


Case  IV.  (§  13). 
a  =  47.55,  c  =  58.40. 
First  find  (as  in  §  13)  log  a 
and  log  h. 

\ogF  =  log<x  +  log  ^  +  colog  2 
log  a     =1.67715 
logb     =1.53015 
colog  2  =  9.69897  — 10 

log  i^  =2.90627 
F         =805.88 


Exercise  IX. 

Solve  the  following  triangles,  finding  the  angles  to  the 
nearest  minute : 


1 

Given. 

Required. 

a  =  6, 

c  =  12. 

^=30°, 

5  =  60°, 

6  =  10.392. 

2 

A  =60°, 

6  =  4. 

5  =  30°, 

c  =  8. 

a  =  6.9282. 

3 

A  =  30°, 

a  =  3. 

5  =  60°, 

c  =  6, 

6  =  5.1961. 

4 

a  =  4, 

6  =  4. 

A  =  B=46° 

,c  =  5.6568. 

5 

a  =  2, 

c  =  2.82843. 

^  =  5  =  45^ 

,6  =  2. 

6 

c  =  627, 

^=23°  30'. 

5  =  66°  30', 

a  =  250.02, 

6  =  575.0. 

7 

c  =  2280, 

^  =  28°  5'. 

5  =  61°  55', 

a  =  1073.3, 

6  =  2011.5. 

8 

c  =  72.15, 

^  =  39°  34'. 

5  =  50°  26', 

a  =  45. 958, 

6  =  55.620. 

9 

c  =  l, 

A  =  36°. 

5  =  54°, 

a  =  0.58779, 

6  =  0.80902. 

10 

c  =  200, 

5  =  21°  47'. 

^=68°  13', 

a  =  185.72, 

6  =  74.22. 

11 

c  =  93.4, 

J5=76°25'. 

^  =  13°  35', 

a  =  21.936. 

6  =  90.788. 

12 

a  =  637, 

A=   4°  35'. 

5  =  85°  25', 

6  =  7946, 

c  =  7971.5. 

13 

a  =  48. 532 

,^  =  36°  44'. 

5  =  53°  16', 

6  =  65.031, 

c  =  81.144. 

14 

a  =  0.0008 

,  A  =  86°. 

5=  4°, 

6  =  0.000055S 

,c  =  0.000802. 

15 

6  =  50.937 

,  B  =  43°48'. 

^=46°  12', 

a  =  53. 116, 

c  =  73.59. 

16 

6  =  2, 

B=   3°  38'. 

^  =  86°  22', 

a  =  31.496, 

c  =  31.559. 

28 


TRIGONOMETRY. 


17 

GIVEN 

REQUIRED, 

a  =  992, 

B=76°W. 

^  =  i3°4r, 

6  =  4074.5, 

c  =  4193.5. 

18 

a  =  73, 

B=6S°52\ 

^1=21°  8', 

6  =  188.86, 

c  =  202,47. 

19 

a  =  2. 189, 

J5  =  45°25'. 

^=44°  35', 

6  =  2.2211, 

c  =  3.1185. 

20 

6  =  4, 

A  =  37°  56'. 

^  =  52°  4', 

a  =  3.1176, 

c  =  5.0714. 

21 

c  =  8590. 

a  =  4476. 

J.  =31°  24', 

^  =  58°  36', 

6  =  7332.8. 

22 

c  =  86.53, 

a  =  71.78. 

^=56°  3', 

2?  =  33°  57', 

6  =  48.324. 

23 

c  =  9.35. 

a  =  8.49. 

^=65°  14', 

5  =  24°  46', 

6  =  3.917. 

24 

c  =  2194, 

6=1312.7. 

^=53°  15', 

5  =  36°  45', 

a  =  1758. 

25 

c  =  30.69. 

6  =  18.256. 

^=53°  30', 

5  =  36°  30', 

a  =  24.67. 

26 

a  =  38.313, 

6=19.522. 

^=63°, 

5  =  27°, 

c  =  43. 

27 

a  =  1.2291. 

6=14.950. 

A=  4°  42', 

5  =  85°  18', 

c=15. 

28 

a  =  415.38, 

6  =  62.080. 

^=81°  30', 

B=  8°  30', 

c  =  420. 

29 

a  =  13.690, 

6  =  16.926. 

^=38°  58', 

5  =  51°  2', 

c  =  21.769. 

30 

c  =  91.92, 

a  =  2.19. 

A=  1°22', 

5  =  88°  38', 

6  =  91.894. 

Compute  the  unknown  parts  and  also  the  area,  having  given : 


31.    a  =  5, 


b  =  6. 


32.    a  =  0M5,c  =  70. 

s 


—  33.   Z»=V2, 

34.  a  =  7, 

35.  b=12, 


c  =  V3. 
A  =  18°  14'. 
A  =  29°  8'. 


36. 
37. 
38. 
39. 
40. 


c  =  68, 
c  =  27, 
a  =47, 


A  =  69' 
B  =  U' 

B  =  4:S' 

B  =  3^° 


54'. 
4'. 
49'. 
44'. 


c  =  8.462, 5  =  86°  4'. 


—  41.  Find  the  value  of  F  in  terms  of  c  and  A. 

--^  42.  Find  the  value  of  F  in  terms  of  a  and  A. 

43.  Find  the  value  of  F  in  terms  of  b  and  A. 

"        44.  Find  the  value  of  F  in  terms  of  a  and  c. 
— ^45.  Given  i^  =  58,    a  =  10  ;  solve  the  triangle. 
.    46.  Given  i^=  18,     b  =  5;  solve  the  triangle. 

47.  Given  i^  =  12,   ^  =  29°;  solve  the  triangle. 

48.  Given  i^=  100,  c  =  22;  solve  the  triangle. 

49.  Find  the  angles  of  a  right  triangle  if  the  hypotenuse  is 
equal  to  three  times  one  of  the  legs. 


THE    RIGHT    TRIANGLE.  29 

"•    50.   Find  the  legs  of  a  right  triangle  if  the  hypotenuse  =  6, 
and  one  angle  is  twice  the  other. 

51.  In  a  right  triangle  given  c,  and  A  =  nB\  find  a  and  h. 

52.  In  a  right  triangle  the  difference  between  the  hypote- 
nuse and  the  greater  leg  is  equal  to  the  difference  between 
the  two  legs  ;  find  the  angles. 


The  angle  of  elevation  of  an  object  (or  angle  of  depression, 
if  the  object  is  below  the  level  of  the  observer)  is  the  angle 
which  a  line  from  the  eye  to  the  object  makes  with  a  horizon- 
tal line  in  the  same  vertical  plane. 

53.  At  a  horizontal  distance  of  120  feet  from  the  foot  of  a 
steeple,  the  angle  of  elevation  of  the  top  was  found  to  be 
60°  30' ;  find  the  height  of  the  steeple. 

'  54.  From  the  top  of  a  rock  that  rises  vertically  326  feet  out 
of  the  water,  the  angle  of  depression  of  a  boat  was  found  to  be 
24° ;  find  the  distance  of  the  boat  from  the  foot  of  the  rock. 

55.  How  far  is  a  monument,  in  a  level  plain,  from  the  eye, 
if  the  height  of  the  monument  is  200  feet  and  the  angle  of 
elevation  of  the  top  3°  30'  ? 

<  56.  In  order  to  find  the  breadth  of  a  river  a  distance  AB 
is  measured  along  the  bank,  the  point  A  being  directly  op- 
posite a  tree  C  on  the  other  side.  The  angle  ABC  i^  also 
measured.  If  AB  is  96  feet,  and  ABC  is  21°  14'  find  the 
breadth  of  the  river. 

If  ABC  were  45°,  what  would  be  the  breadth  of  the  river  ? 

57.  Find  the  angle  of  elevation  of  the  sun  when  a  tower 
a  feet  high  casts  a  horizontal  shadow  h  feet  long.  Find  the 
angle  when  a  =  120,  b  =  70. 

(\  58.  How  high  is  a  tree  that  casts  a  horizontal  shadow  b  feet 
in  length  when  the  angle  of  elevation  of  the  sun  is  A°  ?  Find 
the  height  of  the  tree  Avhen  h  =  80,  A  =  50°. 


30  TRIGONOMETRY. 

59.  What  is  the  angle  of  elevation  of  an  inclined  plane  if  it 
rises  1  foot  in  a  horizontal  distance  of  40  feet  ? 

^  60.  A  ship  is  sailing  due  north-east  with  a  velocity  of  10 
miles  an  hour.  Find  the  rate  at  which  she  is  moving  due 
north,  and  also  due  east. 

61.  In  front  of  a  window  20  feet  high  is  a  flower-bed  6  feet 
wide.  How  long  must  a  ladder  be  to  reach  from  the  edge  of 
the  bed  to  the  window  ? 

62.  A  ladder  40  feet  long  may  be  so  placed  that  it  will  reach 
a  window  33  feet  high  on  one  side  of  the  street,  and  by  turn- 
ing it  over  without  moving  its  foot  it  will  reach  a  window  21 
feet  high  on  the  other  side.     Find  the  breadth  of  the  street. 

63.  From  the  top  of  a  hill  the  angles  of  depression  of  two 
successive  milestones,  on  a  straight  level  road  leading  to  the 
hill,  are  observed  to  be  5°  and  15°.    Find  the  height  of  the  hill. 

0  64.  A  fort  stands  on  a  horizontal  plain.  The  angle  of 
elevation  at  a  certain  point  on  the  plain  is  30°,  and  at  a  point 
100  feet  nearer  the  fort  it  is  45°.     How  high  is  the  fort  ? 

65.  From  a  certain  point  on  the  ground  the  angles  of  eleva- 
tion of  the  belfry  of  a  church  and  of  the  top  of  the  steeple 
were  found  to  be  40°  and  51°  respectively.  From  a  point  300 
feet  farther  off,  on  a  horizontal  line,  the  angle  of  elevation  of 
the  top  of  the  steeple  is  found  to  be  33°  45'.  Find  the 
distance  from  the  belfry  to  the  top  of  the  steeple. 

66.  The  angle  of  elevation  of  the  top  C  of  an  inaccessible 
fort  observed  from  a  point  A,  is  12°.  At  a  point  B,  219  feet 
from  A  and  on  a  line  AB  perpendicular  to  AC,  the  angle  ABC 
is  61°  45'.     Find  the  height  of  the  fort. 


THE    ISOSCELES    TRIANGLE. 


31 


§  15.     The  Isosceles  Triangle. 


An  isosceles  triangle  is  divided  by  the  perpendicular  from 
the  vertex  to  the  base  into  two  equal  right  triangles. 

Therefore,  an  isosceles  triangle  is  determined  by  any  two 
parts  that  determine  one  of  these  right  triangles. 

Let  the  parts  of  an  isosceles  triangle  ABC  (Fig.  18),  among 
which   the   altitude    CD  is   to   be   in- 
cluded, be  denoted  as  follows  : 


a  =  one  of  the  equal  sides, 

c  =  the  base, 

A  =  the  altitude, 

A  =  one  of  the  equal  angles, 

C  =  the  angle  at  the  vertex. 

For  example :    Given  a  and  c ;    le-  A 
quired  A,  C,  h. 


He    D       ^c 
Fig.  18. 


1.     C0SJ=  — : 


G 

2a' 


2.  C  +  2A  =  180°',  .-.  C  =  1S0°-2A  =  2(90°-A). 

3.  h  may  be  found  by  any  one  of  the  equations : 


^2  /i 

h^-\-  —  =  a^:     -  =  sin^; 
4  a 


— -  =tan^; 
4^ 


whence      h=^y  (a  —  ic)  (a  +  i^)  ;   =(ismA-j   =JctanA 


The  area  F  of  the  triangle  may  be  found,  when  c  and  h  are 
given  or  have  been  computed,  by  means  of  the  formula 


F=lch. 


32  TRIGONOMETRY. 


Exercise  X. 

Solve  the  following  isosceles  triangles,  finding  tlie  angles 
to  the  nearest  second  : 

1.  Given  a  and  A ;  find  C,  c,  h. 

2.  Given  a  and  C ;  find  A,  c,  h. 

3.  Given  c  and  A ;  find  C,  a,  h. 

4.  Given  c  and  C ;  find  A,  a,  h. 

5.  Given  h  and  A ;  find  C,  a,  c. 

6.  Given  h  and  C ;  find  J^,  a,  c. 

7.  Given  a  and  h ;  find  ^,  C,  c. 

8.  Given  c  and  A;  find^,  C,  a. 

^^  9.  Given  a  =  14.3,       c  =  11 ;  find  A,  C,  h, 

10.  Given  6^  =  0.295,    ^  =  68°  10';  find  c,  h,F. 

-^11.  Given  c  =  2.352,    (7=69°  49';  find  a,  h,F, 

12.  Given  h  =  7.4847,  ^  =  76°  14' ;  find  a,   c,  F. 

^13.  Given  «  =  6.71,      A  =  6.60;       find^,  C,  c. 

14.  Given  c  =  9,  A  =  20;  find^,  C,  a. 

-15.  Given  c  =  147,       i^=  2572.5;  find^,  C,  a,  h. 

16.  Given  A  =  16.8,      i^^=  43.68;     ^ndA,  C,  a,  e. 

^    17.  Find  the  value  of  F  in  terms  of  a  and  c. 

18.  Find  the  value  of  F  in  terms  of  a  and  C. 

19.  Find  the  value  of  F  in  terms  of  a  and  A 

20.  Find  the  value  of  F  in  terms  of  h  and  (7. 

->'  21.  A  barn  is  40  X  80  feet,  the  pitch  of  the  roof  is  45°; 
find  the  length  of  the  rafters  and  the  area  of  both  sides  of  the 
roof. 

0    22.    In  a  unit  circle  what  is  the  length  of  the  chord  corre- 
sponding to  the  angle  45°  at  the  centre  ? 
o  23.    If  the  radius  of  a  circle  is  30,  and  the  length  of  a  chord 
is  44,  find  the  angle  at  the  centre. 

;.  24.    Find  the  radius  of  a  circle  if  a  chord  whose  length  is 
5  subtends  at  the  centre  an  angle  of  133°. 


THE    REGULAR    POLYGON. 


33 


25.    What  is  the  angle  at  the  centre  of  a  circle  if  the  corre- 
sponding chord  is  equal  to  f  of  the  radius  ? 
V  26.    Find  the  area  of  a  circular  sector  if  the  radius  of  the 
circle  =  12,  and  the  angle  of  the  sector  =  30°. 


i 


§  16.     The  Regular  Polygon. 

Lines  drawn  from  the  centre  of  a  regular  polygon  (Fig.  19) 
to  the  vertices  are  radii  of  the  circumscribed  circle;  and  lines 
drawn  from  the  centre  to  the  middle  points  of  the  sides  are 
radii  of  the  inscribed  circle.  These  lines  divide  the  polygon 
into  equal  right  triangles.  Therefore,  a  regular  polygon  is 
determined  by  a  right  triangle  whose  sides  are  the  radius  of 
the  circumscribed  circle,  the  radius  of  the  inscribed  circle,  and 
half  of  one  side  of  the  polygon. 

If  the  polygon  has  n  sides,  the  angle  of  this  right  triangle 
at  the  centre  is  equal  to 


1  /360°\  180° 

2\-^)      ^^     ^- 


If,  also,  a  side  of  the  polygon,  or  one  of  the  above-men- 
tioned radii,  is  given,  this  triangle  may  be  solved,  and  the 
solution  gives  the  unknown  parts  of  the  polygon. 
Let, 
71  =  number  of  sides, 
c  =  length  of  one  side, 
r  =  radius  of  circumscribed  circle, 
h  =  radius  of  inscribed  circle, 
p  =  the  perimeter, 
i^=the  area. 

Then,  by  Geometry, 

F  =  ^hp,  Fig.  19. 


34  TRIGONOMETRY, 

Exercise  XI. 

~-  1.  Given  n  =  10,    c  =  l;     find  r,   h,  F. 

2.  Given  71  =  12,  ^  =  70;  find  r,    h,  F. 

^  3.  Given  w  =  18,   r  =  1 ;     find  h,  p,  F. 

4.  Given  w  =  20,   r  =  20;  find  h,  c,    F. 

~^  5.  Given  n  =  S,     h  =  l;     find  r,    c,    F. 

6.  Given  71  =  11,  7^=20;  find  r,    h,  c. 

~^7.  Given  w  =  7,    F=7;     find  r,    h,  p. 

8.  Find  the  side  of  a  regular  decagon  inscribed  in  a  unit 
circle.  I^  W  .^  \(^ 

~~    9.   Find  the  side  of  a  regular  decagon  circumscribed  about 
a  unit  circle. 

10.  If  the  side  of  an  inscribed  regular  hexagon  is  equal  to 
1,  find  the  side  of  an  inscribed  regular  dodecagon. 

11.  Given  n  and  c,  and  let  b  denote  the  side  of  the  inscribed 
regular  polygon  having  27i  sides;  find  b  in  terms  of  Tiand  c. 

12.  Compute  the  difference  between  the  areas  of  a  regular 
octagon  and  a  regular  nonagon  if  the  perimeter  of  each  is  16. 

13.  Compute  the  difference  between  the  perimeters  of  a 
regular  pentagon  and  a  regular  hexagon  if  the  area  of  each  is 
12.      N-il 

14.  From  a  square  whose  side  is  equal  to  1  the  corners  are 
cut  away  so  that  a  regular  octagon  is  left.  Find  the  area  of 
this  octagon. 

15.  Find  the  area  of  a  regular  pentagon  if  its  diagonals  are 
each  equal  to  12.         "t  m  <  (^  "^ 

16.  The  area  of  an  inscribed  regular  pentagon  is  331.8; 
find  the  area  of  a  regular  polygon  of  11  sides  inscribed  in 
the  same  circle. 


THE    REGULAR    POLYGON.  35 

17.  The  perimeter  of  an  equilateral  triangle  is  20 ;  find  the 
area  of  the  inscribed  circle.         H .  i  1  h 

18.  The  area  of  a  regular  polygon  of  16  sides,  inscribed  in 
a  circle,  is  100;  find  the  area  of  a  regular  polygon  of  15  sides, 
inscribed  in  the  same  circle. 

19.  A  regular  dodecagon  is  circumscribed  about  a  circle, 
the  circumference  of  which  is  equal  to  1 ;  find  the  perimeter 
of  the  dodecagon.         /•  ^  "^   "3  J^ 

20.  The  area  of  a  regular  polygon  of  25  sides  is  equal  to 
40;  find  the  area  of  the  ring  comprised  between  the  circum- 
ferences of  the  inscribed  and  the  circumscribed  circles. 


CHAPTER   III. 


GONIOMETRY. 


§  17.     Definition  of  Goniometry. 

In  order  to  prepare  the  way  for  the  solution  of  an  oblique 
triangle,  we  now  proceed  to  extend  the  definitions  of  the 
trigonometric  functions  to  angles  of  all  magnitudes,  and  to 
deduce  certain  useful  relations  of  the  functions  of  diiferent 
angles. 

That  branch  of  Trigonometry  which  treats  of  trigono- 
metric functions  in  general,  and  of  their  relations,  is  called 
Goniometry. 

§  18.     Angles  of  any  Magnitude. 

Let  the  radius  OP  of  a  circle  (Fig.  20)  generate  an  angle  by 
turning  about  the  centre  0.     This  angle  will  be  measured  by 

the  arc  described  by  the  point  P; 
and  it  may  have  any  magnitude, 
because  the  arc  described  by  P 
may  have  any  magnitude. 

Let  the  horizontal  line  OA  be 
the  initial  position  of  OP,  and 
let  OP  revolve  in  the  direction 
shown  by  the  arrow,  or  opposite 
to  the  way 'clock  hands  revolve. 
Let,  also,  the  four  quadrants  into 
which  the  circle  is  divided  by  the 
horizontal  and  vertical  diameters 
AA\  BB'j  be  numbered  I.,  II.,  III.,  IV.,  in  the  direction  of  the 
motion. 


GONIOMETRT.  *  37 

During  one  revolution  OF  will  form  with  OA  all  angles  from 
0°  to  360°.  Any  particular  angle  is  said  to  be  an  angle  of  the 
quadrant  in  which  OP  lies;  so  that, 

Angles  between      0°  and    90°  are  angles  of  Quadrant  I. 

Angles  between    90°  and  180°  are  angles  of  Quadrant  II. 

Angles  between  180°  and  270°  are  angles  of  Quadrant  III. 

Angles  between  270°  and  360°  are  angles  of  Quadrant  IV. 

If  OP  make  another  revolution,  it  will  describe  all  angles 
from  360°  to  720°,  and  so  on. 

If  OP,  instead  of  making  another  revolution  in  the  direc- 
tion of  the  arrow,  be  supposed  to  revolve  backwards  about  0, 
this  backward  motion  tends  to  undo,  or  cancel,  the  original 
forward  motion.  Hence,  the  angle  thus  generated  must  be 
regarded  as  a  negative  angle;  and  this  negative  angle  may, 
obviously,  have  any  magnitude.  Thus  we  arrive  at  the  con- 
ception of  an  angle  of  any  magnitude,  positive  or  negative. 


§  19.     General  Definitions  of  the  Functions. 

The  definitions  of  the  trigonometric  functions  may  be 
extended  to  all  angles,  by  making  the  functions  of  any  angle 
equal  to  the  line  values  in  a  unit  circle  drawn  for  the  angle 
in  question,  as  explained  in  §  4.  But  the  lines  that  represent 
the  sine,  cosine,  tangent,  and  cotangent  must  he  regarded  as 
negative,  if  they  are  opposite  in  direction  to  the  lines  that  repre- 
sent the  corresponding  functions  of  an  angle  in  the  first  quad- 
rant; and  the  lines  that  represent  the  secant  and  cosecant  must 
be  regarded  as  negative,  if  they  are  opposite  in  direction  to  the 
moving  radius. 

Figs.  21-24  show  the  functions  drawn  for  an  angle  ^OP  in 
each  quadrant,  taken  in  order.  In  constructing  them,  it  must 
be  remembered  that  the  tangents  to  the  circle  are  always 
drawn  through  A  and  B,  never  through  A'  or  B'. 

Let  the  angle  AOP  be  denoted  by  x ;  then,  in  each  figure, 


38 


TRIGONOMETRY. 


the  absolute  values   of   the    functions  (that   is,  their  values 
without  regard  to  the  signs  +  and  —  )  are  as  follows  : 

Bm.x=MP,         t2inx  =  AT,  secx=  OT, 

cos  X  =  OMj         cot  03  =  BSf  CSC  x  =  OS. 

B  c  ^  B 


Fig.  23. 


Fig.  24. 


Keeping  in  mind  the  position  of  the  points  A  and  B,  we  may 
define  in  words  the  first  four  functions  of  the  angle  x  thus : 
sin  a?  =     the  vertical  projection  of  the  moving  radius; 
cosic=     the  horizontal  projection  of  the  moving  radius; 

the  distance  measured  along  a  tangent  to  the  circle 
tana;  =  ^       from  the  beginning  of  the  first  quadrant  to  the 
moving  radius  produced; 


'-{ 


Kj\ 


60NI0METRY.  39 

r  the  distance  measured  along  a  tangent  to  the  circle 
cotx  ^<      from  the  end  of  the  first  quadrant  to  the  moving 

L     radius  produced. 
Secx  and  esc  a:  are  the  distances  from  the  centre  of  the 
circle  measured  along  the   moving  radius  produced  to  the 
tangent  and  cotangent,  respectively. 

§  20.     Algebraic  Signs  of  the  Functions. 

The  lengths  of  the  lines,  defined  above  as  the  functions  of 
any  angle,  are  expressed  numerically  in  terms  of  the  radius 
of  the  circle  as  the  unit.  But,  before  these  lengths  can  be 
treated  as  algebraic  quantities,  they  must  have  the  sign  +  or 
—  prefixed,  according  to  the  condition  stated  in  §  19. 

The  reason  for  this  condition  lies  in  that  fundamental 
relation  between  algebraic  and  geometric  magnitudes,  in  virtue 
of  which  contrary  signs  in  Algebra  correspond  to  opposite 
directions  in  Geometry. 

The  sine  MP  and  the  tangent  AT  always  extend  from  the 
horizontal  diameter,  but  sometimes  upwards  ai^i  sometimes 
downwards ;  the  cosine  OM  and  the  cotangent  BS  always 
extend  from  the  vertical  diameter,  but  sometimes  towards  the 
right  and  sometimes  towards  the  left.  The  functions  of  an 
angle  in  the  first  quadrant  are  assumed  to  be  positive.  There- 
fore, 

1.  Sines  and  tangents  extending  from  the  horizontal 
diameter  upwards,  are  positive ;  downwards,  negative. 

2.  Cosines  and  cotangents  extending  from  the  vertical 
diameter  towards  the  right,  are  positive ;  towards  the  left^  are 
negative. 

The  signs  of  the  secant  and  cosecant  are  always  made  to 
agree  with  those  of  the  cosine  and  sine,  respectively.  This 
agreement  is  secured  if  secants  and  cosecants  extending  from 
the  centre,  in  the  direction  of  the  moving  radius,  are  considered 
positive  ;  in  the  opposite  direction,  negative. 


40 


TRIGONOMETRY. 


Hence,  the  signs  of  the  functions  for  each  quadrant  are : 

In  Quadrant  I.  all  the  functions  are  positive. 

In  Quadrant  II.  the  sine  and  cosecant  only  are  positive. 

In  Quadrant  III.  the  tangent  and  cotangent  onlysiTe  positive. 

In  Quadrant  IV.  the  cosine  and  secant  only  are  positive. 

§  21.    Functions  of  a  Variable  Angle. 

Let  the  angle  x  increase  continuously  from  0°  to  360°; 
what  changes  will  the  values  of  its  functions  undergo? 

It  is  easy,  by  reference  to  Fig.  25,  to  trace  these  changes 
throughout  all  the  quadrants. 


Fig.  25. 


1.  The  Sine.  In  the  first  quadrant,  the  sine  MP  increases 
from  0  to  1;  in  the  second  it  remains  positive,  and  decreases 
from  1  to  0;  in  the  third  it  is  negative,  and  increases  in 
absolute  value  from  0  to  1 ;  in  the  fourth  it  is  negative,  and 
decreases  in  absolute  value  from  1  to  0. 


GONIOMETRY.  41 

2.  The  Cosine.  In  the  first  quadrant,  the  cosine  OM  de- 
creases from  1  to  0 ;  in  the  second  it  becomes  negative,  and 
increases  in  absolute  value  from  0  to  1;  in  the  third  it  is 
negative,  and  decreases  in  absolute  value  from  1  to  0 ;  in  the 
fourth  it  is  positive,  and  increases  from  0  to  1. 

3.  The  Tangent.  In  the  first  quadrant,  the  tangent  AT 
increases  from  0  to  oc  ;  in  the  second  quadrant,  as  soon  as  the 
angle  exceeds  90°  by  the  smallest  conceivable  amount,  the 
moving  radius  OF',  prolonged  in  the  direction  opposite  to  that 
of  OF',  will  cut  AT  Sit  Q,  point  T'  situated  very  far  below  A; 
hence,  the  tangents  of  angles  near  90°  in  the  second  quadrant 
have  very  large  negative  values.  As  the  angle  increases,  the 
tangent  AT'  continues  negative,  but  diminishes  in  absolute 
value.  When  x  =  180°,  then  T'  coincides  with  A,  and  tan  180° 
=  0.  In  the  third  quadrant,  the  tangent  is  positive,  and 
increases  from  0  to  oo  ;  in  the  fourth  it  is  negative,  and 
decreases  in  absolute  value  from  O)  to  0. 

4.  The  Cotangent  In  the  first  quadrant,  the  cotangent  ^^S^ 
decreases  from  oo  to  0;  in  the  second  quadrant  it  is  negative, 
and  increases  in  absolute  value  from  0  to  oo ;  in  the  third  and 
fourth  quadrants  it  has  the  same  sign,  and  undergoes  the  same 
changes  as  in  the  first  and  second  quadrants,  respectively. 

5.  The  Secant.  In  the  first  quadrant,  the  secant  OT  in- 
creases from  1  to  GO ;  in  the  second  it  is  negative  (being 
measured  in  the  direction  opposite  to  that  of  OF'),  and 
decreases  in  absolute  value  from  oo  to  1;  in  the  third  it 
is  negative,  and  increases  in  absolute  value  from  1  to  oo ;  in 
the  fourth  it  is  positive,  and  decreases  from  go  to  1^ 

6.  The  Cosecant.  In  the  first  quadrant,  the  cosecant  OS 
decreases  from  c»  to  1 ;  in  the  second  it  is  positive,  and 
increases  from  1  to  go  ;  in  the  third  it  is  negative,  and 
decreases  in  absolute  value  from  oo  to  1  ;  in  the  fourth  it 
is  negative,  and  increases  in  absolute  value  from  1  to  go. 


42 


TRIGONOMETRY. 


The  limiting  values  of  the  functions  are  as  follows : 


Sine 

0° 

90' 

180- 

270° 

360- 

±0 

1 

±0 

-1 

±0 

Cosine 

1 

±0 

-1 

±0 

1 

Tangent 

±0 

±cc 

±0 

±  00 

±0 

Cotangent     .... 

±00 

±0 

d=  CC 

-±o 

d=oo 

Secant   

1 

±0) 

-1 

rtoo 

1 

Cosecant   

dboo 

1 

=b  00 

-  1 

iO) 

Sines  and  cosines  extend  from  -|-1  to  —1;  tangents  and 
cotangents  from  -f-  ^  to  —  oo ;  secants  and  cosecants  from 
-f-  00  to  -\-l,  and  from  —  1  to  —  oo. 

In  the  table  given  above  the  double  sign  ±  is  placed  before  0  and  oo. 
From  the  preceding  investigation  it  appears  that  the  functions  always 
change  sign  in  passing  through  0  and  oo ;  and  the  sign  +  or  —  prefixed 
to  0  or  00  simply  shows  the  direction  from  which  the  value  is  reached. 

Take,  for  example,  tan  90°  :  The  nearer  an  acute  angle  is  to  90°,  the 
greater  the  positive  value  of  its  tangent ;  and  the  nearer  an  obtuse  angle 
is  to  90°,  the  greater  the  negative  value  of  its  tangent.  When  the  angle 
is  90°,  OP  (Fig.  25)  is  parallel  to  A  T,  and  cannot  meet  it.  But  tan  90° 
may  be  regarded  as  extending  either  in  the  positive  or  in  the  negative 
direction;  and  according  to  the  view  taken,  it  will  be  +  oo  or  —  oo. 


§  22.    Functions  of  Angles  Larger  than  360°. 

The  functions  of  360° +  x  are  the  same  in  sign  and  in 
absolute  value  as  those  of  x ;  for  the  moving  radius  has  the 
same  position  in  both  cases.     If  w  is  a  positive  integer, 

The  functions  of  (n  X  360°  +  x)  are  the  same  as  those  of  x. 

For  example:  The  functions  of  2200°  (6  X  360° -f  40°)  are 
equal  to  the  functions  of  40°. 


goniometrt.  43 

§  23.    Extension  of  Formulas  [1]-[3]  to  all  Angles. 

The  Formulas  established  for  acute  angles  in  §  6  hold  true 
for  all  angles.     Thus,  Formula  [1], 

sin^x  +  cos^x  =  1, 

is  universally  true ;  for,  whether  MP  and  OM  (Fig.  25)  are 
positive  or  negative,  MP^  and  OM^  are  always  positive,  and 
in  each  quadrant  JZp'+  057'=  (jF  =  l. 

Also,  r21     tan  x  = j 

•■  -■  cos  a: 

{sin  X  X  CSC  x^l, 
cos  a-  X  sec  a:  =  1, 
tan  X  X  cot  x  =  l, 

are  universally  true ;  for  they  are  in  harmony  with  the  alge- 
braic signs  of  the  functions,  given  at  the  end  of  §  20 ;  and  we 
have  in  each  quadrant  from  the  similar  triangles  OMP,  OAT, 
OBS,  (Fig.  25)  the  proportions 

AT  :  OA  =  MP:  OM, 
MP:  OP=OB  :  OSy 
OM:  OP=OA  :  OT, 
AT  :  OA=OB  :  BS, 

which,  by  substituting  1  for  the  radius,  and  the  right  names 
for  the  other  lines,  are  easily  reduced  to  the  above  formulas. 
Formulas  [l]-[3]  enable  us,  from  a  given  value  of  one 
function,  to  find  the  absolute  values  of  the  other  five  func- 
tions, and  also  the  sign  of  the  reciprocal  function.  But  in 
order  to  determine  the  proper  signs  to  be  placed  before  the 
other  four  functions,  we  must  know  the  quadrant  to  which 
the  angle  in  question  belongs ;  or  the  sign  of  any  one  of  these 
four  functions ;  for,  by  §  20,  it  will  be  seen  that  the  signs 
of  any  two  functions  that  are  not  reciprocals  determine  the 
quadrant  to  which  the  angle  belongs. 


44  TRIGONOMETRY. 

Example.  Given  sin  cc  =  -f  |,  and  tan  x  negative  ;  find  the 
values  of  the  other  functions. 

Since  sin  x  is  positive,  x  must  be  an  angle  in  Quadrant  I.  or 
II. ;  but,  since  tan  x  is  negative.  Quadrant  I.  is  inadmissible. 


By  [1],  cosa^  =  ±Vl-i|  =  ±t- 

Since  the  angle  is  in  Quadrant  II.  the  minus  sign  must  be 
taken,  and  we  have 

cos  ic  =  —  f . 
By  [2]  and  [3], 

tanx^  —  I,     cotx  =  — f,     seca:  =  — |,     cscic:=|. 

Exercise  XII. 

1.  Construct  the  functions  of  an  angle  in  Quadrant  II. 
What  are  their  signs  ? 

2.  Construct  the  functions  of  an  angle  in  Quadrant  III. 
What  are  their  signs  ? 

3.  Construct  the  functions  of  an  angle  in  Quadrant  IV. 
What  are  their  signs  ? 

4.  What  are  the  signs  of  the  functions  of  the  following 
angles :   340°,  239°,  145°,  400°,  700°,  1200°,  3800°  ? 

5.  How  many  angles  less  than  360°  have  the  value  of  the 
sine  equal  to+  f,  and  in  what  quadrants  do  they  lie  ? 

6.  How  many  values  less  than  720°  can  the  angle  x  have 
if  cos  a;:=  +  f,  and  in  what  quadrants  do  they  lie  ? 

7.  If  we  take  into  account  only  angles  less  than  180°,  how 
many  values  can  x  have  if  sin  a;  =  f  ?  if  cos  x=^\  ?  if  cos  x  = 
—  I?  iftana;  =  |?  if  cota!  =  — 7? 

8.  Within  wliat  limits  must  the  angle  x  lie  if  cos  cc  =  —  |  ? 
ifcot(r  =  4?  ifsec2c  =  80?  ifcscx  =  — 3?  (if  a^  <  360°). 

9.  In  what  quadrant  does  an  angle  lie  if  sine  and  cosine 
are  both  negative  ?  if  cosine  and  tangent  are  both  negative  ? 
if  the  cotangent  is  positive  and  the  sine  negative  ? 


GONIOMETRT.  45 

10.  Between  0°  and  3600°  how  many  angles  are  there  whose 
sines  have  the  absolute  value  f  ?  Of  these  sines  how  many- 
are  positive  and  how  many  negative  ? 

11.  In  finding  cos  cc  by  means  of  the  equation  cosaj  = 
±  V  1  —  sui^x,  when  must  we  choose  the  positive  sign  and 
when  the  negative  sign  ? 

12.  Given  cos  a:  =  —  V  ^  ;  find  the  other  functions  when 
X  is  an  angle  in  Quadrant  II. 

13.  Given  tan  ic  =  V  3  ;  find  the  other  functions  when  x  is 
an  angle  in  Quadrant  III. 

14.  Given  sec  x  =  -\-7,  and  tan  x  negative ;  find  the  other 
functions  of  x. 

15.  Given  cot  x  =  —  3 ;  find  all  the  possible  values  of  the 
other  functians. 

16.  What  functions  of  an  angle  of  a  triangle  may  be  nega- 
tive ?    In  what  case  are  they  negative  ? 

17.  What  functions  of  an  angle  of  a  triangle  determine  the 
angle,  and  what  functions  fail  to  do  so  ? 

18.  Why  may  cot  360°  be  considered  equal  either  .to  +  oo 
or  to  —  00  ? 

19.  Obtain  by  means  of  Formulas  [l]-[3]  the  other  func- 
tions of  the  angles  given  : 

(i.)  tan    90°  =  00.  (iii.)  cot  270°  =  0. 

(ii.)  cos  180°  =  —  1.  (iv.)  esc  360°  =  —  oo. 

20.  Find  the  values  of  sin  450°,  tan  540°,  cos  630°,  cot  720°, 
sin  810°,  CSC  900°. 

21.  For  what  angle  in  each  quadrant  are  the  absolute  values 
of  the  sine  and  cosine  equal  ? 

Compute  the  values  of  the  following  expressions : 

22.  a  sin  0°-\-b  cos  90°  —  c  tan  180°. 

23.  a  cos  90°  —  b  tan  180°  +  c  cot  90°. 

24.  a  sin  90°  —  b  cos  360°  -\-(a  —  b)  cos  180°. 

25.  (a^  —  b^)  cos  360°  —  Aab  sin  270°. 


46 


TRIGONOMETRY. 


§  24.     Reduction  of  Functions  to  the  First  Quadrant. 

In  a  unit  circle  (Fig.  26)  draw  two  diameters  PR  and  QS 

equally  inclined  to  the  horizon- 
tal diameter  AA',  or  so  that  the 
angles  AOP,  A'OQ,  A'OH,  and 
AGS  shall  be  equal.  From  the 
points  F,  Q,  B,  S  let  fall  perpen- 
diculars to  AA^ ;  the  four  right 
triangles  thus  formed,  with  a 
common  vertex  at  0,  are  equal; 
because  they  have  equal  hypote- 
nuses (radii  of  the  circle)  and 
equal  acute  angles  at  0.  There- 
fore, the  perpendiculars  PM,  QN,  EN,  SM,  are  equal.  Now 
these  four  lines  are  the  sines  of  the  angles  AOP,  AOQ,  AOR, 
and  AOS,  respectively.     Therefore,  in  absolute  value, 

BuiAOP  =  ^mAOQ  =  &inAOR  =  ^mAOS. 

And  from  §  23  it  follows  that  in  absolute  value  the  cosines 
of  these  angles  are  also  equal ;  and  likewise  the  tangents,  the 
cotangents,  the  secants,  and  the  cosecants.* 

Hence,  for  every  acute  angle  (AOP)  there  is  an  angle  in  each 
of  the  higher  quadrants  whose  functions,  in  absolute  value,  are 
equal  to  those  of  this  acute  angle. 

Let  ZAOP  =  x,/_  POP  =  y  ;    then  x-\-y  =  90°,  and  the 
functions  of  x  are  equal  to  the  co-named  functions  of  y  (§  5)  ; 
and     Z  ^0^  (in  Quadrant  11.)    =180°  — :r=   90°  +  ?/, 
Z.AOR  (in  Quadrant  III.)  ==  180° -^x  =  270°  —  y, 
Z.AOS  (in  Quadrant  IV.)  =  360°  —  x  =  270°  +  y. 

Hence,  prefixing  the  proper  sign  (§  20),  we  have : 

*  In  future,  secants,  cosecants,  versed  sines,  and  coversed  sines  will  be 
disregarded.  Secants  and  cosecants  may  be  found  by  [3],  versed  sines 
and  coversed  sines  by  VII.  and  VIII.,  page  5,  if  wanted,  but  they  are 
seldom  used  in  computations. 


GONIOMETRT.  47 

Angle  in  Quadrant  II. 

sin  (180°  —  ic)  =      sin  x.  sin  (90°  +  ?/)  =     cos  y. 

cos  (180°  —  if)  =  —  cos  X.  cos  (90°  +  y)  =  —  sin  y. 

tan  (180°  —  x)  =  —  tan  x.  tan  (90°  +  y)  =  —  cot  y. 

cot  (180°  —  x)  =  —  cot  X.  cot  (90°  +  2/)  =  —  tan  y. 

Angle  in  Quadrant  HI. 

sin  (180°  -\~x)  =  —  sin  ic.  sin  (270°  —  ?/)=--  cos  y. 

cos  (180°  -^x)  =  —  cos  cc.  cos  (270°  —  y)  =  —  sin  y. 

tan  (180°  +  x)  =      tan  x.  tan  (270°  —  v/)  =      cot  y. 

cot  (180°  +  x)  =      cot  a;.  cot  (270°  —  y)=      tan  y. 

Angle  in  Quadrant  IV. 

sin  (360°  —  x)  =  —  sin  x.  sin  (270°  -\-y)=  —  cos  y. 

cos  (360°  —  x)=      cos  a-.  cos  (270°  +  ij)  =      sin  2/. 

tan  (360°  —  x)=  —  tan  a;.  tan  (270°  -f-  ?/)  =  —  cot  2/. 

cot  (360°  —  x)=  —  cot  a:.  cot  (270°  +  ?/)=  —  tan  3/. 

Remark.  The  tangents  and  cotangents  may  be  found  directly  from 
the  figure,  or  by  formula  [2]. 

It  is  evident  from  these  formulas, 

1.  The  functions  of  all  angles  can  be  reduced  to  the  functions 
of  angles  not  greater  than  45°. 

2.  If  an  acute  angle  he  added  to  or  subtracted  from  180°  or 
360°,  the  functions  of  the  resulting  angle  are  equal  in  absolute 
value  to  the  like-named  functions  of  the  acute  angle  ;  but  if  an 
acute  angle  be  added  to  or  subtracted  from  90°  or  270°,  the  func- 
tions of  the  resulting  angle  are  equal  in  absolute  value  to  the 
co-named  functions  of  the  acute  angle. 

3.  A  given  value  of  a  sine  or  cosecant  determines  two  supple- 
mentary angles,  one  acute,  the  other  obtuse  ;  a  given  value  of  any 
other  function  determines  only  one  angle:  acute  if  the  value  is 
positive,  obtuse  if  the  value  is  negative.  [See  functions  of 
(180° -x).] 


'    f 


48 


TRIGONOMETRY. 


§  25.    Angles  whose  Difference  is  90°. 

The  general  form  of  two  such  angles  is  x  and  90°  +  ^j  and 
they  must  lie  in  adjoining  quadrants.     The  relations  between 

their  functions  were  found  in  §  24, 
but  only  for  the  case  when  x  is 
acute.  These  relations,  however, 
may  be  shown  to  hold  true  for  all 
values  of  x. 

In  a  unit  circle  (Fig.  27)  draw 
two  diameters  FB  and  QS  per- 
pendicular to  each  other,  and  let 
fall  to  AA'  the  perpendiculars 
FM,  QH,  RK,  and  SN.  The 
right  triangles  OMF,  OHQ,  OKR, 
and  ONS  are  equal,  because  they  have  equal  hypotenuses 
and  equal  acute  angles  FOM,  OQH,  ROK,  and  OSN. 

Therefore,  OM=QH=OK=NS, 

and  FM=OH  =  KR=ON. 

Hence,  taking  into  account  the  algebraic  sign, 
sin^O^=      cos^OP;   ^inAOS  =      cos AOR 

cos  AOQ  =  — sin  AOF;  cos  AOS  =  —  sinAOR: 

sin  AOR  =      cos  AOQ;    sin  (360°  +  ^ OP)  =      cos^O^; 
cos^OP  =  — sin^O^;    cos  (360°  +  ^ OP)  =  —  sin  ^ 0^. 
In  all  these  equations,  if  x  denote  the  angle  on  the  right- 
hand  side,  the  angle  on  the  left-hand  side  will  be  90° +  x. 
Therefore,  if  x  be  an  angle  in  any  one  of  the  four  quadrants, 

sin  (90°  -\-x)=      cos  x,  tan  (90°  -f  cc)  =  —  cot  x, 

cos  (90°  -\-x)  =  —  sin  x,  cot  (90°  -]-x)  =  —  tan  x. 

In  like  manner,  it  can  be  shown  that  all  the  formulas  of 

§  24  hold  true,  whatever  be  the  values  of  the  angles  x  and  y. 

Hence,  in  every  case  the  algebraic  sign  of  the  function  of  the 

resulting  angle  will  he  the  same  as  when  x  and  y  are  both  acute. 


(fl/- 


goniometrt.  49 

§  26.    Functions  of  a  Negative  Angle. 

If  the  angle  AOF  (Fig.  26)  is  denoted  by  x,  the  equal  angles 
AOS,  generated  by  a  backward  rotation  of  the  moving  radius 
from  the  initial  position  OA,  will  be  denoted  by  —  x.  It  is 
obvious  that  the  position  OS  of  the  moving  radius  for  this 
angle  is  identical  with  its  position  for  the  angle  360°  — x. 
Therefore,  the  functions  of  the  angle  —x  are  the  same  as 
those  of  the  angle  360°  —  x ;  or  (§  24), 

sin  ( —  x)  =  —  sin  cc,  tan  ( —  x)  =  —  tan  x, 

cos  (—  x)  =      cos  X,  cot  (—  x)  =  —  cot  X. 

Exercise  XIII. 

1.  Express  sin  250°  in  terms  of  the  functions  of  an  acute 
angle  less  than  45°. 

Ans.    sin  250°  =  sin  (270°  —  20°)  =  -  cos  20°. 

Express  the  following  functions  in  terms  of  the  functions 
of  angles  less  than  45°  : 

2.  sin  172°.  8.  sin  204°.  14.  sin  163°  49'. 

3.  cos  100°.  9.  cos  359°.  15.  cos  195°  33'. 

4.  tan  125°.  10.  tan  300°.  16.  tan  269°  15'. 

5.  cot    91°.  11.  cot  264°.  17.  cot  139°  17'. 

6.  sec  110°.  12.  sec  244°.  18.  sec  299°  45'. 

7.  CSC  157°.  13.  CSC  271°.  19.  esc    92°  25'. 

Express  all  the  functions  of  the  following  negative  angles 
in  terms  of  those  of  positive  angles  less  than  45°  : 

20.  -75°.  22.    -200°.  24.   -52°  37'. 

21.  —127°.  23.    —345°.  25.   —196°  54'. 

26.    Find  the  functions  of  120°. 

Hint.       120°  -  180°-  60°,  or,  120°  =  90°  +  30° ;  then  apply  §  24. 


60  TRIGONOMETRY. 

Find  the  functions  of  the  following  angles  : 

27.  135°.  29.   210°.  31.    240°.  33.    —30°. 

28.  150°.  30.    225°.  32.    300°.  34.    —225°. 

35.  Given  sin  x  =  —  V-J-,  and  cos  x  negative  ;  find  the  other 
functions  of  x,  and  the  value  of  x. 

36.  Given  cota:;  =  — V3,  and  x  in  Quadrant  II.;  ^nd  the 
other  functions  of  x,  and  the  value  of  x. 

37.  Find  the  functions  of  3540°. 

38.  What  angles  less  than  360°  have  a  sine  equal  to  —  ^  ? 
a  tangent  equal  to  — V3  ? 

39.  Which  of  the  angles  mentioned  in  Examples  27-34 
have  a  cosine  equal  to  —  V-J  ?  a  cotangent  equal  to  —  V3  ? 

40.  What  values  of  x  between  0°  and  720°  will  satisfy  the 
equation  sin  x  =  +  ^  ? 

41.  Find  the  other  angle  between  0°  and  360°  for  which  the 
corresponding  function  (sign  included)  has  the  same  value  as 
sin  12°,  cos  26°,  tan  45°,  cot  72°,  sin  191°,  cos  120°,  tan  244°,cot  357°. 

42.  Given  tan  238°  =  1.6 ;  find  sin  122°. 

43.  Given  cos  333°  =  0.89 ;  find  tan  117°. 

Simplify  the  following  expressions : 

44.  acos(90°  — ^)  +  ^>cos(90°  +  a^). 

45.  m  cos  (90°  — x)  sin  (90°- x). 

46.  {a  —  b)  tan  (90°  -x)-\-  (a  +  b)  cot  (90°  +  x). 

47.  a^-\-P  —  2abcos(lS0°  —  x). 

48.  sin  (90°  +  x)  sin  (180°  +  ^)  +  cos  (90°  +  x)  cos  (180°  —  x). 

49.  cos(180°+cc)cos(270°— ?/)  — sin(180°+cc)sin(270°-?/). 

50.  tanicH-tan(— ?/)  — tan(180°  — y). 

51.  For  what  values  of  x  is  the  expression  sincc  +  cosa? 
positive,  and  for  what  values  negative  ?  Eepresent  the  result 
by  shading  the  sectors  corresponding  to  the  negative  values. 

52.  Answer  the  question  of  last  example  for  sin  x  —  cos  x. 

53.  Find  the  functions  of  (x  —  90°)  in  functions  of  x. 

54.  Find  the  functions  of  (x  — 180°)  in  functions  of  x. 


GONIOMETRY. 


51 


§  27.    Functions  of  the  Sum  of  Two  Angles. 

In  a  unit  circle  (Fig.  28)  let  the  angle  AOB^=x,  the  angle 
BOC=y',  then  the  angle  AOC  = 
x-\-y. 

In  order  to  express  sin  {x  -\-  y) 
and  cos  (x  -\-  y)  in  terms  of  the 
sines  and  cosines  of  x  and  y,  draw 
CFA_OA,  CD1_0B,  DE^OA, 
DG±  CF;  then  CD  =  smy,  OB 
=  cos?/,  and  the  angle  BCG  = 
the  an^lc  6^Z>0  =  ic.     Also, 

sin  (x  +  yy=  CF=  I)F-\-  CG. 

BE 

——  =  sm  X ;     hence,  BF  =  sin  x  X  OB  =  sin  x  cos  y. 

CG 


CB 


=  cos  X ;     hence,  CG  =  cos  x  X  CB  =  cos  x  sin  y. 


m 


Therefore,  sin  (x  +  y)  =  sin  x  cos  y  +  cos  x  sin  y . 

Again,  cos  (x  +  y)  =  OF=  OF  —BG. 

OF 

-—  =  cos  X ;     hence,  OF  =  cos  x  X  OB  =  cos  x  cos  y. 

DC 

-—-  =  sin  X  ;     hence,  BG  =  since  X  CB  =  sin  x  sin y. 
uB 

Therefore,  cos  (x  +  y)  =  cos  x  cos  y  —  sin  x  sin  y .  [5] 

In  this  proof  x  and  y,  and  also  the  sum  x-\-y,  are  assumed 
to  be  acute  angles.  If  the  sum 
x-{-y  ot  the  acute  angles  x  and 
y  is  obtuse,  as  in  Fig.  29,  the 
proof  remains,  word  for  word, 
the  same  as  above,  the  only  dif- 
ference being  that  the  sign  of 
OF  will  be  negative,  as  BG  is 
now  greater  than  OF.  The  above  formulas^  therefore,  hold 
true  for  all  acute  angles  x  and  y. 


52  TRIGONOMETRY. 

If  these  formulas  hold  true  for  any  two  acute  angles  x  and 
y,  they  hold  true  when  one  of  the  angles  is  increased  by  90°. 
Thus,  if  for  x  we  write  cc'  =  90°  -|-  x,  then,  by  §  25, 

sin  (x'  -\-y)  =  sin  (90°  -\-x-{-y)=      cos  (x  -\-  ?/), 
cos  (x'  -J-  ?/)  =  cos  (90°  +  ic  +  2/)  =  —  sin  (x  -{-  y). 

Hence,  by  [5],  sin  (x'  -[-  2/)  =      cos  x  cos  y  —  sin  x  sin  ?/, 
by  [4],  cos  (x'  +  y)  =  —  sin  x  cos  y  —  cos  x  sin  y, 

Now,  by  §  25,  cos x  =      sin (90° -\-x)=      sin x\ 
sin  cc  =  —  cos  (90°  -\-x)  =  —  cos  x\ 

Substitute  these  values  of  cos  x  and  sin  x,  then 

sin  (x^  -\-y^=z  sin  £c'  cos  y  -\-  cos  x^  sin  y, 
cos  (cc'  +  y)  =  cos  £c'  cos  y  —  sin  ic'"sin  y. 

It  follows  that  Formulas  [4]  and  [5]  hold  true  if  either 
angle  is  repeatedly  increased  by  90°  ;  therefore  they  apply  to 
all  angles  whatever. 

By  §  23, 

,       ,     ,     .       sin  (x  -f  y)      sin  x  cos  y  +  cos  x  sin  ?/ 

tan  (x-^y)  = ) — p^  = '^—^ — : ^' 

cos  {X  -\-  y)      cos  X  cos  ?/  —  sm  x  sm  y 

If  we  divide  each  term  of  the  numerator  and  denominator 
of  the  last  fraction  by  cos  x  cos  y,  and  again  apj)ly  §  23,  we 
obtain 

,      /     ,     .        tanx+tany  ^^^ 

*^°^^+y)=l-tanxtany'  ^ 

In  like  manner,  by  dividing  each  term  of  the  numerator 
and  denominator  of  the  value  of  cot  (x  -\-  t/)  by  sin  x  sin  y,  we 
obtain 

....      cotxcoty— 1 
cot(x  +  y)  =  — - — r-^4 —  r71 

^    ^^^       coty  +  cotx  "-'J 


GONIOMETRY. 


53 


§  28.    Functions  of  the  Difference  of  Two  Angles. 

Iji  a  unit  circle  (Fig.  30)  let  tlie  angle  AOB  =  x,  COB  =  y, 
then  the  angle  AOC  =  x  —  ?/. 

In  order  to  express  sin  (x  —  y) 

and   cos  (x  —  ?/)    in   terms    of   the 

sines  and  cosines  of  x  and  y,  draw 

CF  _L  OA,   CD  ±  OB,  DE  J_  OA, 

DG  _L  FC  prolonged;  then  CD= 

sin  y,    OD  =  cos  y,  and   the   angle 

I)CG  =  the  angle  FDC=x.     And, 

sin  (x  —  y)=  CF=DE  —  CG. 

^         '^^  Fig.  30. 

DE 

-r-r  =  sin  X ;     hence,  DE  =  sin  x  X  OD  =  sin  x  cos  y. 

CC 

— —  =  cos  X ;     hence,  CG  =  cos  x  X  CD  =  cos  x  sin  y. 

Therefore,     sin(x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y.  [8] 

Again,  cos(x  —  y)=  OF=OE -\-DG. 

OE 
OD 
DG 


=  cos  x ;     hence,  OE  =  cos  x  X  OD  =  cos  x  cos  y. 

hence,  DG  =  sin  x  X  -CD  =  sin  x  sin  y. 
Therefore,     cos  (x  —  y)  =  cos  x  cos  y  +  sin  x  sin  y .  [9] 


^  =  sm^;_ 


In  this  proof,  both  x  and  y  are  assumed  to  be  acute  angles ; 
but,  whatever  be  the  values  of  x  and  y,  the  same  method  of 
proof  will  always  lead  to  Formulas  [8]  and  [9],  when  due 
regard  is  paid  to  the  algebraic  signs. 

The  general  application  of  these  formulas  may  be  at  once 
shown  by  deducing  them  from  the  general  formulas  estab- 
lished in  §  27,  as  follows  : 

It  is  obvious  that  (x  —  y)-\-y  =  X'  If  we  apply  Formulas 
[4]  and  [5]  to  (x  —  y)-\-y,  then 


64  TRIGONOMETRY. 

sin  J  (a?  —  y)-\-yl  or  sin  x  =  sin  (x  —  ?/)  cos  y  +  cos  (x  —  ?/)  sin  y, 
cos  \(x  —  y)-{-yi  or  cos x  =  cos (x  —  y)  cos y  —  sin  (x  —  y) sin y. 

Multiply  the  first  equation  by  cos  y,  the  second  by  sin  j/, 

sin  X  cos  y=  *  sin  (x  —  ?/)  cos^y  +  cos  (x  —  y)  sin y  cos  2^, 
cos  a;  sin  y  =  —  sin  (x  —  y)  sin'^y  -\-  cos  (x  —  y)  sin  y  cos  y ; 

whence,  by  subtraction, 

sin  X  cos  ?/  —  cos  x  sin  ?/  =  sin  (x  —  y)  (sin^y  +  cos^y). 

But  sin^y  +  cos^^  =  1 5  therefore,  by  transposing, 

sin  (x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y. 

Again,  if  we  multiply  the  first  equation  by  sin  y,  the  second 
equation  by  cos  y,  and  add  the  results,  we  obtain,  by  reducing, 

cos  (x  —  y)  =  cos  x  cos  y  -}-  sin  x  sin  y. 

Therefore,  Formulas  [8]  and  [9],  like  [4]  and  [5],  from 
which  they  have  been  derived,  are  universally  true. 
From  [8]  and  [9],  by  proceeding  as  in  §  27,  we  obtain 

.      .         -       tanx  — tany  ^_^ 

*'^°("-y)=l  +  tanxtany-  t^^^ 

,  .         .      cotxcoty+1*  ^^^^ 

Formulas  [4]-[ll]  may  be  combined  as  follows: 
sin  (xdzy)  =  sin  x  cos  ?/  d=  cos  x  sin  ?/, 
cos  (x  ±  ?/)  =  cos  cc  cos  y :+:  siu  a^  sin  y, 
tan  a;  ±  tan  y 


tan  (a;  ±  ?/) 


tan  x  tan  ?/ 


^  ,         ,       cot  x  cot  ?/  =F  1 

cot  (a!  ±  ?/)  = --^—^ — 

cot  y  ±  cot  x 


goniometry.  55 

§  29.    Functions  of  Twice  an  Angle. 
If  y  =  x,  Formulas  [4]-[7],  become 

sin  2  X  =  2  sin  X  cos  X.    [12]         cos  2  x  =  cos^x  —  sin^x.  [13] 
.      _  2tanx  ,. ,_,  ._         cot^x— 1  _^_ 

By  these  formulas  the  functions   of  twice  an  angle  are 
found  when  the  functions  of  the  angle  are  given. 

§  30.    Functions  of  Half  an  Angle. 

Take  the  formulas 

cos^a;  +  sin^j;  =  1  [1  ] 

cos^aj  —  sin^x  =  cos  2x       ~  [1 3] 

Subtract,  2  sin^j?  =  1  —  cos  2  x 

Add,  2  cos^^  =  1  +  cos  2  x 

Whence 


/l  — cos2a^  /l  +  . 

=  ±\ ,      cosa;  =  ±^— ^ 


cos  2  X 
sm  X  "  .    -  .      . 


2  ~  >/  2 

These  values,  if  z  is  put  for  2x,  and  hence  ^;^  for  x,  become 

•    1          ,  ^/l  — cosz    ^,„^                 ,                /l  +  cosz     ^^„^ 
siniz  =  it:^ [16]  cosiz=:±^ — — [17] 

Hence,  by  division  (§  23), 

ta„iz  =  ±Vf^^'  [18]  cotiz=±Vi±^.   [19] 

^l  +  cosz    •-     -"  ^1  — cosz    *-     -* 

By  these  formulas  the  functions  of  half  an  angle  may  be 
computed  when  the  cosine  of  the  entire  angle  is  given. 

The  proper  sign  to  be  placed  before  the  root  in  each  case 
depends  on  the  quadrant  in  which  the  angle  -J  z  lies.     (§  21.) 

Let  the  student  show  from  Formula  [18]  that 


tan  I-  ^  =  \  — — •    (See  page  22,  Note.) 


56  tkigonometrt. 

§  31.    Sums  and  Differences  of  Eunctions. 
From  [4],  [5],  [8],  and  [9],  by  addition  and  subtraction : 
sin  (ic  +  ?/)  +  sin  (x—  y)=^      2  sin x  cos  i/, 
sin  (x-{-  y)  —  sin  (x  —  ?/)  =      2  cos  x  sin  y, 
cos  (x-\-y)-{-  cos  (x  —  y)=^      2  cos  x  cos  y, 
cos  (x-{-y)  —  cos  (x  —  y)  =  —  2  sin  x  sin  ?/ ; 

or,  by  making  £c  +  ?/  =  ^,  and     x  —  y^=B, 

and  therefore,  x^=^{A-\- B),    and     ?/  =  |- (y1  —  B), 

sinA+sinB=:     2siiii(A  +  B)cosi-(Ar-B).  [20] 

sin  A  — sin  B=     2cosi(A  +  B)sini(A— B).  [21] 

cos  A  +  cos  B  =     2  cos  ^  (A  +  B)  cos  i  (A  —  B).  [22] 

cosA  — cosB=-2sinJ(A  +  B)sini(A  — B).  [23] 

From  [20]  and  [21],  by  division,  we  obtain 

sin^  +  sin^  i  /  ^   i   t>n     4.1/1       t>\ 

— — ^ — -  =  tan \(A-\-B) cot  1  (A  —  B)  : 

sm^  — sm^  2v      I      y        2v  y> 

or,  since  cot -^(-4 — B)^= 


tan  ^(yl—i?) 

sin  A  -[-  sin  B  _  tan  j^  (A  +  B) 
sin  A  —  sin  B      tan  ^  (A  —  B) 

Exercise  XIV. 


[24] 


1.  Find  the  value  of  sin  (x  -\-  y)  and  cos  (x  -j-  y),  when  sin  x 
=  f,  cosa;  =  f,  smy  =  j%  cos?/  =  f|. 

2.  Find  sin  (90°  — 3/)  and  cos(90°  — ?/)  by  making  0^  =  90° 
in  Formulas  [8]  and  [9]. 

Find,  by  Formulas  [4:]-[ll],  the  first  four  functions  of: 

3.  90° +  2/.  8.    360° -y.  13.    -y. 

4.  180°-?/.  9.    360°  +  ?/.  14.   45°  —  ?/. 

5.  180°  +  ?/.  10.    ic  — 90°.  15.    45°+?/. 

6.  270°-?/.  11.   0^-180°.  16.    30°  +  ?/. 

7.  270°  +  ?/.  12.   X  — 270°.  17.    60°  —  ?/. 


GONIOMETRT.  67 

18.  Find  sinSic  in  terms  of  since. 

19.  Find  cos  3  a?  in  terms  of  cos  x. 

20.  Given  tan  ^cc  =  1 ;  find  cos  x. 

21.  Given  cot-J-ic=:  V3;  find  sin  a?. 

22.  Given  since  =  0.2  ;  find  sin^cc  and  cos^-cc. 

23.  Given  cos  x  =  0.5  ;  find  cos  2x  and  tan  2ic. 

24.  Given  tan  45°  =  1 ;  find  the  functions  of  22°  30'. 

25.  Given  sin  30°  =  0.5 ;  find  the  functions  of  15°. 

sin  33°  + sin  3°     ., 


26.    Prove  that  tan  18°  = 


cos  33°  +  cos  3*^ 


Prove  the  following  formulas  : 

1  ^  nrr  n  2tanx  _^    ^      .  sin  a; 

^-^  27.    sin2a;  =  ^_^^_,^'  29.    tan  Jo;  = 


l  +  tan^oj  '  i_|_cosi» 

oo  o         1  ~  tan^a;  _ .         .  ^  sin  x 

28.    cos2a;  =  — -rr — 5—  30.    coti-a;  =  :; 

1  +  tannic  ^         1  —  cos  x 

31.  sin  ^x  =b  cos  -J-a:  =  Vl  ±  sin  x. 

__.     tana^rttany 

32.  — 7 —^  =  =t  tan  x  tan  y. 

cot  X  ±  cot  y 

33.  tan(45°-a^)  =  iq^^^- 

^  ^      1  +  tan  X 

If  Aj  Bj  C  are  the  angles  of  a  triangle,  prove  that : 

34.  sin^  +  sin^  +  sin  (7==4cos-j-^  cos^i?cos-|-C. 

35.  cos ^  +  cos  J5  +  cos  C  =  l+4sin^^  sin ^^  sin -^  C. 

36.  tan^  +  tan B  +  tan  C  =  t^nA  X  tan^  X  tan  C. 

37.  cot  |^  +  cot  i^  +  cot  ^C  =  Got^AX  cot  ^B  X  cot  1  C. 

Change  to  forms  more  convenient  for  logarithmic  computa- 
tion : 

38.  cot  a;  +  tan  aj.  43.  1  +  tan  a;  tan  y. 
^  39.  cot  X  —  tan  x.  44.  1  —  tan  x  tan  y. 
_    40.    cot  a?  +  tan  2/.                            45.    cota;  cot^z  +  l. 

41.   cot  a?  —  tany.  46.    cot  a?  cot  y  —  1. 

^     1  —  cos  2a?  tan  a:  +  tan  2/ 

14- cos  2a;  '    cot  a;  +  cot  1/ 


58  trigonometry. 

§  32.     Anti-Trigonometric  Functions. 

If  y  is  any  trigonometric  function  of  an  angle  x,  then  x  is 
said  to  be  the  corresponding  anti-trigonometric  function  of  y. 
Thus,  if  2/  =  sin  x,  x  is  the  anti-sine  of  y,  or  inverse  sine 
of  y.     The  anti-trigonometric  functions  of  y  are  written 
sin-^y,  tan-^?/,  sec~^2/j  vers"^?/, 

cos~^2/,  cot~^?/,  csc~^?/,  covers"^  y. 

These  are  read,  the  angle  whose  sine  is  y,  etc. 
For  example,  sin  30°  =  ^;    hence  30°  =  sin"' ^.      Similarly 

90°  =  cos-^  0  =:  sin-i  ;i^ .  ^nd  45°  =  tan-^  1  =  sin-^  —-= ;  etc. 

V2 

The  symbol  -^  must  not  be  confused  with  the  exponent  —  1.     Thus 

sin-ix  is  a  very  different  expression  from  - — >  which  would  be  written 

(sinx)— 1.  On  the  Continent  of  Europe  mathematical  writers  employ  the 
notation  arc  sin,  arc  cos,  etc. ,  for  sin— i,  cos-i,  etc.  But  the  latter  symbols 
are  most  common  in  England  and  America. 

There  is  an  important  difference  between  the  trigonometric 
and  the  anti-trigonometric  functions.  When  an  angle  is  given, 
its  functions  are  all  completely  determined;  but  when  one 
of  the  functions  is  given  the  angle  may  have  any  one  of  an 
indefinite  number  of  values.  Thus,  if  sin  ?/=  |,  y  may  be  30°, 
or  150°,  or  either  of  these  increased  or  diminished  by  any 
integral  multiple  of  360°  or  2'jr,  but  cannot  take  any  other 
values.  Accordingly  sin~^  ^ = 30°  ±  2  mr,  or  150°  =b  2  titt,  where 
n  is  any  positive  integer.  Similarly,  tan~^l  =  45°d=27i7r  or 
225°  +  2  WTT ;  i.e.,  tan-^  1  =  45°  ±  utt. 

Since  one  of  the  angles  whose  sine  is  x  and  one  of  the  angles 
whose  cosine  is  x  together  make  90°,  and  since  similar  rela- 
tions hold  for  the  tangent  and  cotangent,  for  the  secant  and 
cosecant,  and  for  the  versed  sine  and  coversed  sine,  we  have 

sin~^  x  -\-  cos~^  ^  =  o '  sec~^  ic  +  csc~^  ^  =  o ' 


GONIOMETRY.  69 

tan~^  X  -\-  cot~^  ^  =  9 '  vers~^  x  -\-  covers  "^  cc  =  - , 

where  it  must  be  understood  that  each  equation  is  true  only 
for  a  particular  choice  of  the  various  possible  values  of  the 
functions.  For  example,  if  x  is  positive,  and  if  the  angles 
are  always  taken  in  the  first  quadrant,  the  equations  are 
correct. 

Exercise  XV. 

1.  Find  all  the  values  of  the  following  functions : 
sin-iiV3,  tan-i^Va,  vers-^^,  cos-i(— iV2),  csc-i(V2), 
tan-i  Qc^    ggg-i  2^    cos-^  (—  \  V3) . 

2.  Prove  that  sin~^(—£c)=—sin-^  a;;  cos~^(— ic)=7r— cos~^ic. 
^     3.    If  sin~^x -|-  sin"^?/  =  tt,  prove  that  x^y. 

^    4.    If  2/  =  sin~^-J,  find  tan  3/. 

5.  Prove  that  cos  (sin-^ic)  =  Vl  —  x^. 

6.  Prove  that  cos  (2  sin~^  cc)  =  1  —  2  icl 

X  ~T~  II 

7.  Prove  that  tan  (tan~^  x  +  tan-^  ?/)  = -- 

^  ^       1  —  xy 

8.  If  ic=  \l^,  find  all  the  values  of  sin~^ic  +  cos"~^a;. 

9.  Prove  that  tan~^  I    ,  \  =  sin~^a;. 

10.  Find  the  value  of  sin  (tan~\\). 

11.  Find  the  value  of  cot  (2  sin-if ). 

12.  Find  the  value  of  sin  (tan-^^  +  tan"^^). 

13.  If  sin~^ic  =  2  cos~^x,  find  x. 

2x  t 

14.  Prove  that  tan  (2  tan"^  x)  =    _    g- 

_L         X 

2x 

15.  Prove  that  sin  (2  tan"^  x)  =  ■^- 


CHAPTER   IV. 

THE    OBLIQUE    TRIANGLE. 

§  33.    Law  of  Sines. 

Let  a,  B,  C  denote  the  angles  of  a  triangle  ABC  (Figs.  31 
and  32),  and  a,  b,  c,  respectively,  the  lengths  of  the  opposite 
sides. 

Draw  CD  _L  AB,  and  meeting  AB  (Fig.  31)  or  AB  pro- 
duced (Fig.  32)  at  D.     Let  CD  =  h. 


C 


A           c 

Fig. 

D 

31. 

B             A~ 

c           B 

Fig.  32. 

In  both  figures, 

^            '       A 
0 

In  Fig.  31, 

h 

-  =  sm  B. 

a 

In  Fig,  32, 

^  =  sin  (180°- 

-B)  =  smB. 

Therefoi-e,  whether  h  lies  within  or  without  the  triangle, 
we  obtain,  by  division, 


a  _  sin  A 
b      sinB 


[25] 


THE    OBLIQUE    TRIANGLE. 


61 


By  drawing  perpendiculars  from  the  vertices  A  and  B  to 
the  opposite  sides  we  may  obtain,  in  the  same  way, 

b sinj5  a sin^ 

c      sin  C 


sm  C 


Hence  the  Law  of  Sines,  which  may  be  thus  stated : 

The  sides  of  a  triangle  are  proportional  to  the  sines  of  the 

opposite  angles. 

If  we  regard  these  three  equations  as  proportions,  and  take 

them  by  alternation,  it  will  be  evident  that  they  may  be 

written  in  the  symmetrical  form, 

a     h     c 

sin  A      sin  ^      sin  G 

Each  of  these  equal  ratios  has  a  simple  geometrical  meaning 
which  will  appear  if  the  Law  of  Sines  is  proved  as  follows  : 

Circumscribe  a  circle  about  the  triangle  ABC  (Fig.  33), 
and  draw  the  radii  OA,  OB,  OC] 
these  radii  divide  the  triangle  into 
three  isosceles  triangles.  Let  B 
denote  the  radius.  Draw  031 
J_BC.  By  Geometry,  the  angle 
BOC  =  2  A;  hence,  the  angle 
BOM=A,  then  BM=EsmBOM 
=  BsinA. 

.-.BCoT  a  =  2BsmA. 

In  like  manner,  h  =  2  B  sin  B, 
and  G  =  2  B  sin  C.  Whence  we 
obtain 

a  h 


Fig.  33. 


2B 


sinvl      sin^      sin  (7 


That  is  :  The  ratio  of  any  side  of  a  triangle  to  the  sine  of  the 
opposite  angle  is  numerically  equal  to  the  diameter  of  the  cir- 
cumscribed circle. 


62  trigonometry. 

§  34.   Law  of  Cosines. 

This  law  gives  the  value  of  one  side  of  a  triangle  in  terms 
of  the  other  two  sides  and  the  angle  included  between  them. 

In  rigs.  31  and  32,        w"  =  h'  +  RD\ 

In  Fig.  31,  BB  =c  —  AD', 

in  Fig.  32,  BD  =AD~c ; 

in  both  cases/  BD"  =  AB^  —  2cXAB-\-c\ 

Therefore,  in  all  cases,  a^  =  h^  +  Ajf  -\-c^  —  2cX  AB. 

Now,  h''-}-AF  =  b% 

and  AlB  ^hcosA. 

Therefore,  B,^  =  h^+c^  —  2\)CG0sA.  1262 

In  like  manner,  it  may  be  proved  that 
i2^^2_j_^2  —  2ac  cos  B, 
c^  =  a^-\-P  —  2abG0sC. 

The  three  formulas  have  precisely  the  same  form,  and  the 
law  may  be  stated  as  follows  : 

The  square  of  any  side  of  a  triangle  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides,  dinmiished  by  twice  their 
product  into  the  cosine  of  the  included  angle. 

§  35.    Law  op  Tangents. 
By  §  33,  a  :b  =  sin  A  :  sin  ^ ; 

whence,  by  the  Theory  of  Proportion, 

a  —  b sin  ^  —  sin  B 

a-\-b      sin  ^4"  sin  ^ 

But  by  [24],  page  56, 

sin  A  —  sin  B tan  ^(A  —  B) 

sin  A  +  sin  B      tan  ^(A-\-B) 
Therefore, 

a-b_tani(A-B) 

a  +  b      tan^(A4-B)  L^^-J 


THE    OBLIQUE    TRIANGLE.  63 

By  merely  changing  the  letters, 

a  —  c tan  \{A  —  C)         h  —  c tan  \{B  —  (7) 

a4-c"~tani(^+C)'        H^'~tan^(^+C) 

Hence  the  Law  of  Tangents  : 

The  difference  of  two  sides  of  a  triangle  is  to  their  sum  as  the 
tangent  of  half  the  difference  of  the  opjiosite  angles  is  to  the 
tangent  of  half  their  sum. 

Note.      If  in  [27]  6>  a,  then  B'^A.     The  formula  is  still  true,  but  to 
avoid  negative  numbers,  the  formula  in  this  case  should  be  written 
h  —  a  _  tani(E  — J.) 

Exercise  XVI. 

1.  What  do  the  formulas  of  §  33  become  when  one  of  the 
angles  is  a  right  angle  ? 

2.  Prove  by  means  of  the  Law  of  Sines  that  the  bisector 
of  an  angle  of  a  triangle  divides  the  opposite  side  into  parts 
proportional  to  the  adjacent  sides. 

3.  What  does  Formula  [26]  become  when  A  =  90°  ?  when 
j;  =  0°  ?  when  J.  =  180°  ?  What  does  the  triangle  become  in 
each  of  these  cases  ? 

Note.  The  case  when  A  =  90°  explains  why  the  theorem  of  §  34  is 
sometimes  termed  the  Generalized  Theorem  of  Pythagoras. 

4.  Prove  (Figs.  31  and  32)  that  whether  the  angle  B  is 
acute  or  obtuse,  c  =  a  cos  B-{-h  cos  A.  What  are  the  two  sym- 
metrical formulas  obtained  by  changing  the  letters  ?  What 
does  the  formula  become  when  B  =  90°  ? 

5.  From  the  three  following  equations  (found  in  the  last 
example)  prove  the  theorem  of  §  34 ; 

c:=a  cos  B  -{-h  cos  A, 
h=^a  cos  C  -\-  c  cos  A, 
a=^h  cos  C  -\rG cos B. 

Hint.  Multiply  the  first  equation  by  c,  the  second  by  6,  the  third 
by  a ;  then  from  the  first  subtract  the  sum  of  the  second  and  third. 


64  TRIGONOMETRY. 

6.  In  Eormula  [27]  what  is  the  maximum  value  of  -J  (A — B)  ? 

7.  Find  the  form  to  which  Formula  [27]  reduces,  and 
describe  the  nature  of  the  triangle,  when 

(i.)   (7  =  90°  ;  (ii.)  A  —  B  =  90°,  and  B=a 

§  36.    The  Solution  of  an  Oblique  Triangle. 

The  formulas  established  in  §§  33-35,  together  with  the 
equation  A-\-B-\-C  =  180°,  are  sufficient  for  solving  every 
case  of  an  oblique  triangle.  The  three  parts  that  determine 
an  oblique  triangle  may  be  : 

I.    One  side  and  two  angles ; 
II.    Two  sides  and  the  angle  opposite  to  one  of  these  sides ; 

III.  Two  sides  and  the  included  angle ; 

IV.  The  three  sides. 

Let  A,  B,  C  denote  the  angles,  a,  h,  c  the  sides  respectively. 

§  37.    Case  I. 

Given  one  side  a,  and  two  angles  A  and  B;  find  the  remain- 
ing parts  C,  b,  and  c. 

1.  C=180°-(^  +  J5). 

„     b      smB  ^      asinB         a  .    „ 

2.  -  =  - — -;         .■.b  =  — — -  =  - — rXsm^. 
a      sm^  sm^        sm^ 

^    c       sin  (7  <x  sin  C  a  .     ^ 

6.    -  =  — — - ;         .•.<?  =  — : — —  =  — — 7  X  sm  C. 
a      sm^  sm^        sm^ 

Example,   a  =  24.31,  A  =  45°  18',  B  =  22°  11'. 
The  work  may  be  arranged  as  follows  : 

log  a  =  1.38578  =  1.38578 

colog  sin  A  =  0.14825  =  0.14825 

log  sin  B  =  9.57700    log  sin  C  =  9.96556 

log  c  =  1.49959 

c  =  31.593 


a=: 

24.31     1 

A  = 

45° 

18' 

B  = 

22° 

11' 

-^B  = 

67° 

29' 

C  = 

112° 

31' 

log  J  =  1.11103 
^»  =  12.913 


Note.     When  —  10  is  omitted  after  a  logarithm  or  a  cologarithm,  it 
must  be  remembered  that  the  log  or  the  colog  is  10  too  large. 


THE    OBLIQUE    TRIANGLE.  65 

Exercise  XVII. 

""l.    Given  a  =  500,  ^  =  10°  12',  ^  =  46°  36'; 

find  (7  =  123°  12',  ^»=  2051.48,  c  =  2362.61. 

2.    Given  a  =  795,  ^  =  79°  59',  i?  =  44°41'; 

find  (7  =  55°  20',  ^'=  567.688,  c  =  663.986. 

-,  3.    Given  a  =  804,  A  =  99°  55',  ^  =  45°  1' ; 

find  (7  =  35°  4',  ^»  =  577.313,  c  =  468.933. 

4.    Given  «  =  820,  ^  =  12°  49',  ^  =  141°  59'; 

find  C  =  25°  12',  ^  =  2276.63,  c  =  1573.89. 

_  5.    Given  c  =  1005,  ^  =  78°  19',  ^  =  54°  27'; 

find  C  =  47°  14',  a  =  1340.6,  b  =  1113.8. 

6.    Given  ^»  =  13.57,  i?=13°57',  (7  =  57°13'; 

find  ^  =  108°  50',  a  =  53.276,  c  =  47.324. 

-I  7.    Given  a  =  6412,  A  =  70°  55',  (7= 52°  9' ; 

find  ^  =  56°  56',  ^=5685.9,  c  =  5357.5. 

8.  Given  ^»  =  999,  ^  =  37°  58',  (7=65°  2'; 

find  ^  =  77°,  a  =  630.77,      c  =  929.48. 

9.  In  order  to  determine  the  distance  of  a  hostile  fort  A 
from  a  place  B,  a  line  BC  and  the  angles  ABC  and  BCA 
were  measured,  and  found  to  be  322.55  yards,  60°  34',  and 
56°  10',  respectively.     Find  the  distance  AB. 

10.  In  making  a  survey  by  triangulation,  the  angles  B  and 
(7  of  a  triangle  ABC  were  found  to  be  50°  30'  and  122°  9', 
respectively,  and  the  length  ^C  is  known  to  be  9  miles. 
Find  AB  and  AC. 

11.  Two  observers  5  miles  apart  on  a  plain,  and  facing 
each  other,  find  that  the  angles  of  elevation  of  a  balloon  in 
the  same  vertical  plane  with  themselves  are  55°  and  58°, 
respectively.  Find  the  distance  from  the  balloon  to  each 
observer,  and  also  the  height  of  the  balloon  above  the  plain. 

12.  In  a  parallelogram  given  a  diagonal  d  and  the  angles 
X  and  y  which  this  diagonal  makes  with  the  sides.  Find  the 
sides.     Find  the  sides  if  fZ  =  11.237,  a^  =  19°  1',  and2/  =  42°54'. 


66  TRIGONOMETRY. 

^  13.  A  lighthouse  was  observed  from  a  ship  to  bear  N.  34°  E. ; 
after  the  ship  sailed  due  south  3  miles,  it  bore  N.  23°  E.  Find 
the  distance  from  the  lighthouse  to  the  ship  in  both  positions. 

Note.  The  phrase  to  hear  N.  34°  E.  means  that  the  line  of  sight  to 
the  lighthouse  is  in  the  north-east  quarter  of  the  horizon,  and  makes, 
with  a  line  due  north,  an  angle  of  34°. 

14.  In  a  trapezoid  given  the  parallel  sides  a  and  h,  and  the 
angles  x  and  y  at  the  ends  of  one  of  the  parallel  sides.  Find 
the  non-parallel  sides.  Compute  the  results  when  a  =  15, 
J  =  7,  0^  =  70°,  7/  =  40°. 

Solve  the  following  examples  without  using  logarithms  : 

-V  15.    Given  h  =  7.07107,     A  =  30°,     C  =  105° ;  find  a  and  c. 

16.  Given  c  =  9.562,         ^=45°,     i?  =  60°;     find  a  and  &. 

17.  The  base  of  a  triangle  is  600  feet,  and  the  angles  at  the 
base  are  30°  and  120°.     Find  the  other  sides  and  the  altitude. 

18.  Two  angles  of  a  triangle  are,  the  one  20°,  the  other  40°. 
Find  the  ratio  of  the  opposite  sides. 

19.  The  angles  of  a  triangle  are  as  5  :  10  :  21,  and  the  side 
opposite  the  smallest  angle  is  3.     Find  the  other  sides. 

20.  Given  one  side  of  a  triangle  equal  to  27,  the  adjacent 
angles  equal  each  to  30°.  Find  the  radius  of  the  circum- 
scribed circle.     (See  §  33,  Note.) 

§  38.    Case  XL 

Given  two  sides  a  and  b,  and  the  angle  A  opposite  to  the 
side  a;  find  the  remaining  parts  B,  C,  c. 

This  case,  like  the  preceding  case,  is  solved  by  means  of 
the  Law  of  Sines. 


Q.                        sin  5      5                         .            hs\Y\.A 
bmce  - — 7  = -J   therefore  sinZ?=: 


sin^ 


a 


(7  =  180°  — (^  +  J5). 


THE    OBLIQUE    TRIANGLE. 


67 


.     -    .  c       sm  (7     , ,        o  a  sm  C 

And  since         -^=— — r?   therefore  c  =  —. — —- 
a      sm  A  sm  A 

When  an  angle  is  determined  by  its  sine  it  admits  of  two 
values,  which  are  supplements  of  each  other  (§  24)  ;  hence, 
either  value  of  B  may  be  taken  unless  excluded  by  the  con- 
ditions of  the  problem. 

li  a'>  b,  then  by  Geometry  A'>  B,  and  B  must  be  acute 
whatever  be  the  value  of  A)  for  a  triangle  can  have  only 
one  obtuse  angle.  Hence,  there  is  07ie,  and  only  one^  triangle 
that  will  satisfy  the  given  conditions. 

li  a  =  h,  then  by  Geometry  A  =  B',  both  A  and  B  must  be 
acute,  and  the  required  triangle  is  isosceles. 

If  a<,h,  then  by  Geometry  A<iB,  and  A  must  be  acute 
in  order  that  the  triangle 

may  be  possible.     If  ^  is  C^ 

acute,  it  is  evident  from 
Fig.  34,  where  Z.BAC  =  A, 
AC  =  b,  CB=  CB'  =  a, 
that  the  two  triangles  ACB 
and  ACB'  will  satisfy  the 
given  conditions,  provided 
a  is  greater  than  the  per- 
pendicular CF ;  that  is,  provided  a  is  greater  than  h  sin  A 
(§  11).  The  angles  ABC  smdAB'C  are  supplementary  (since 
Z_ABC  =  /_BB'C)\  they  are  in  fact  the  supplementary 
angles  obtained  from  the  formula 


sin  7? 


h^mA 


If,  however,  a  =  h  ^\nA=CP  (Fig.  34),  then  sin  B  —  l, 
B  =  90°,  and  the  triangle  required  is  a  right  triangle. 

li  a<h  sin  J,  that  is,  <  CF,  then  sin^  >  1,  and  the  tri- 
angle is  impossible. 


68  TRIGONOMETRY. 

These  results,  for  convenience,  may  be  thus  stated : 

Two  solutions  J  if  A  is  acute  and  the  value  of  a  lies  between 
b  and  h  sin  A. 

No  solution ;  if  ^  is  acute  and  <x  <  Z>  sin  A ; 
or  if  A  is  obtuse  and  a  <  Z>. 

One  solution ;  in  all  other  cases. 

The  number  of  solutions  can  often  be  determined  by  inspec- 
tion.    In  case  of  doubt,  find  the  value  of  b  sin  A. 

Or  we  may  proceed  to  compute  log  sin  B.  If  log  sin  ^  =  0, 
the  triangle  required  is  a  right  triangle.  If  log  sin^>0,  the 
triangle  is  impossible.  If  log  sin  i>  <  0,  there  is  one  solution 
when  a'>b;  there  are  two  solutions  when  a<^b. 

When  there  are  two  solutions,  let  B\  C\  c',  denote  the 
unknown  parts  of  the  second  triangle  j  then, 

^'  =  180°-^,     C'  =  lSO°  —  (A-\-B')=B  —  A, 
a  sin  C 


c' 


sin^ 


Examples. 

1.  Given  a  =  16,  b  =  20,  ^==106°;    find   the   remaining 
parts. 

In  this  case  a<6,  and  ^  >90°  ;  therefore  the  triangle  is  impossible. 

2.  Given   a  =  36,    ^^  =  80,    ^  =  30°;    find   the   remaining 
parts. 

Here  we  have  &sin^  =  80  X  ^  =  40 ;  so  that  a<6sin^,  and  the 
triangle  is  impossible. 

3.  Givena  =  72630,  ^»  =  117480,  J  =  80°0'50";  find^,  C,c. 


a  =  72630 
b=  117480 
^  =  80°0'50' 


cologa=  5.13888 

log&=  5.06996 

log  sin  A  =  9.99337 

log  sin  1?=  0.20221 


Here  logsinB>0. 
.-.  no  solution. 


THE    OBLIQUE    TRIANGLE. 


69 


4.    Given  a  =  13.2,  b  =  15.7,  A  =  57°  13'  15" ;  find  B,  C,  c. 


a=  13.2 
6=15.7 
^  =  57°  13' 15" 

Here  log  sin  B  =  0, 
.-.  a  right  triangle. 

cologa  =  8.87943 

log  6=  1.19590 

logsin^  =  9.92467 

log  sin  5  =0.00000 

5=90° 

.-.  C  =  32°46'45' 


c  =  6  cos  J. 

log  6  =1.19590 

logcos^  =  9.73352 

logc  =  0.92942 

c=8.5 


5.    Given  a  =  767,  h  =  2A2,  ^  =  36°  53' 2";  find  B,  C,  c. 


a  =  767 
6=242 
A  =  36°  53'  2' 


Here  a  >  6, 

and  log  sin  5  <;  0. 

.-.  one  solution. 

6.    Given  a 
other  parts. 

a  =177.01 
6  =  216.45 
A  =  35°  36'  20" 


cologa=  7.11520 

log6=  2.38382 

logsinJ.  =  9.77830 

log  sin  J5=  9.27732 

B  =  10°  54'  58' 
.-.  C  =  132°  12'  0' 


loga  =  2.88480 

logsinC  =  9.86970 

colog  sin  A  =  0.22170 

logc  =  2.97620 

c  =  946.675 


=  177.01,  Z*  =  216.45,  ^  =  35°  36' 20";  find  the 


Here  a  <  6, 

and  log  sin  B  <C  0. 

.-.  two  solutions. 


colog  a  =  7.75200 

log6  =  2.33536 

logsin^  =  9.76507 

log  sin  J5=  9.85243 

5  =45°  23' 28' 

or  134° 36' 32 

.-.  C  =  99°  0'  12 

or  9°  47' 8" 

Exercise 


loga  =  2.24800 

cologsinJ.  =  0.23493 

logsin  (7  =9.99462 

logc  =  2.47755 


2.24800 
0.23493 
9.23035 


1.71328 
c=  300.29  or  51.675 


I 

cisE  XVTII.  ' 


1.    Determine   the   number   of   solutions    in    each   of  the 
following  cases : 

(i.)  a  =  m,  5  =  100,  ^  =  30°. 

(ii.)  ci  =  50,  J  =  100,  ^  =  30°.' 

(iii.)  a  =  40,  5  =  100,  ^  =  30°. 

(iv.)  6^  =  13.4,  5  =  11.46,  ^  =  77°  20'. 

(v.)  a  =  70,  5  =  75,  A  =  m\ 

(vi.)  a  =  134.16,  5  =  84.54,  ^  =  52°  9' 11". 

(vii.)  a  =  200,  5  =  100,  ^  =  30°. 


70  TRIGONOMETRY. 

2.  Given  a  =  840,  b  =  AS5,  J  =  21°  31'; 

find  ^  =  12°  13' 34",  (7=  146°  15' 26",    c  =  1272.15. 

3.  Given  a  =  9.399,  ^'  =  9.197,  ^  =  120°  35'; 

find  i?  =  57°  23' 40",  (7  =  2°  1' 20",  c  =  0.38525. 

4.  Given  «  =  91.06,  ^^  =  77.04,  ^  =  51°9'6"; 

find  ^  =  41°  13',  6^  =  87°  37' 54",      c  =  116.82. 

^     5.    Given  a  =  55.55,  h  =  66.66,  ^  =  77°  44' 40"; 

find  A  =  54°  31'  13",  C  =  47°  44'  7",     c  =  50.481. 

Given  «  =  309,  ^^  =  360,  J  =21°  14' 25"; 

findi?  =  24°  57' 54",  (7  =  133°47'41",  c  =  615.67, 
^'=155°  2' 6",    C"=3°43'29",    c'  =  55.41. 

7.  Given  a  =  8.716,  Z'  =  9.787,  ^  =  38°  14' 12"; 

findj5  =  44°l'28",    (7  =  97°  44' 20",  c  =  13.954, 
^'=135°58'32",  C"=5°47'16",    c'=  1.4202. 

8.  Given  a  =  4.4,  ^•  =  5.21,  yi  =  57°  37' 17"; 

findi?  =  90°,  (7=  32°  22' 43",  c  =  2.79. 

9.  Giveni^=34,  S(,=  22,  :i5  =  30°20'; 

find  ^  =  51°  18' 27",    (7  =  98°  21' 33",     c  =  43.098, 
^'  =  128°  41' 33",  (7' =  20°  58' 27",    c'  =  15.593. 

10.  Given  Z^  =  19,  c  =  18,  (7=15°  49'; 

find  i?  =  16°  43' 13",    ^  =  147°  27' 47",  a  =  35.519, 
^'  =  163°  16'  47",  A'  =  0°  54'  13",       a'  =  1.0415. 

11.  Given  a  =  75,  h  =  29,  ^  =  16°  15' 36";  find  the  differ- 
ence between  the  areas  of  the  two  corresponding  triangles 
without  finding  their  areas  separately. 

12.  Given  in  a  parallelogram  the  side  a,  a  diagonal  d, 
and  the  angle  A  made  by  the  two  diagonals ;  find  the  other 
diagonal.     Special  case  :  a  =  35,  d  =  63,  A  =  21°  36'  30". 


THE    OBLIQUE    TRIANGLE. 


71 


§  39.    Case  III. 

Given  two  sides  a  and  h  and  the  included  angle  C ;  find  the 
remaining  parts,  A,  B,  and  c. 

Solution  I.  The  angles  A  and  B  may  both  be  found  by 
means  of  Formula  [27],  §  35,  which  may  be  written 

a  —  h 


tan^(^— 5): 


Xtan|(^  +  ^). 


■  Since  i  (^  +  ^)  =  i  (180*^  —  C),  the  value  of  ^{A-^B)  is 
known ;  so  that  this  equation  enables  us  to  find  the  value  of 
^{A—B).     We  then  have 

and  i(A^B)-i{A  —  B)  =  B. 

After  A  and  B  are  known,  the  side  c  may  be  found  by  the 
Law  of  Sines,  which  gives  its  value  in  two  ways,  as  follows : 

a  sin  C  b  sin  C 

c  =  ^. — 7-'     or    c  =  — ^ — —• 
BYnA  smi/ 

Solution  II.    The  third  side  c  may  be  found  directly  from 
the  equation  (§  34) 

c  =  \lo}  -\-  V^  — 2  ah  cos  C\ 
and  then,  by  the  Law  of  Sines,  the  following  equations  for 
computing  the  values  of  the  angles  A  and  B  are  obtained : 


sin  ^  =  a  X 


sin  G 


smB  =  bX 


sin  C 


Solution  III.  If,  in  the  triangle  ^^C  (Fig.  35),  BD  is  drawn 
perpendicular  to  the  side  AC,  then 

B 


tan  A  =  ——-  = 


BD 


Now 
and 


'.tan^ 


AD      AC  — DC 
BD  =  a  sin  C 
DC=  a  cos  C. 
a  sin  C 


b  —  a  cos  C 


72 


TBIGONOMETBT. 


By  merely  changing  the  letters, 

bsinC 


tan^ 


a  —  b  cos  C 


It  is  not  necessary,  however,  to  use  both  formulas.  When 
one  angle,  as  A,  has  been  found,  the  ol^er,  B,  may  be  found 
from  the  relation  ^  +  ^+  C  =  180°. 

When  the  angles  are  known,  the  third  side  is  found  by  the 
Law  of  ^ines,  as  in  Solution  L 

Note.  When  all  three  unknown  parts  are  required.  Solution  L  is  the 
most  conyenient  in  practice.  When  only  the  third  side  c  is  desired.  Solu- 
tion n.  may  be  used  to  adrantage,  proYided  the  rahies  of  cfi  and  b^  can 
be  readily  obtained  without  the  aid  of  loganthms.  But  Solutions  IL  and 
m.  are  not  adapted  to  logarithmic  woik. 


Given  a  =  748,  6=375,   C  =  63°35'30";   find  A,  B, 
^3  Of 


1. 

and  c, 

a  +  6  =  1123 
a-6=373 

i{A-^B)=  ^SPirW 


A=  86*23'  9" 
5=  30*  r2r' 


log(a-6)=2.57171 

colog(a  +&)=6.94962 

log  tan  i<^+ ^=0.20766 

log  tan  1(^—^=9.72899 

f(^-B)  =  28*10' 54'' 


log6  =  2.57403 

logsm  C  =  9.95214 

cologsin  B  =  0.30073 

logc  =  2.82690 

c=  671.27 


Note.  In  the  above  Example  we  use  the  an^e  B  in  finding  the  side 
c,  ntiber  than  the  angle  Af  because  A  is  near  90*,  and  therefore  the  use 
of  its  sine  should  be  avoided. 

2.   Given  a  =  4,  c  =  6,  J5=60°;  find  the  third  side  b. 

Here  Sc^ntion  IL  may  be  used  to  advantage.     We  have 

6=Vtf2  +  c2  — 2acco«lf  =  Vl6  + 36-24  =  \^; 
Iog28=  1.44716,    log>^  =  0.72358,     V28  =  5.2915; 
thatis,         &  =  5,2915. 


THE    OBLIQl-^    TRIANGLE.  73 


Exercise  XIX. 


^  1.    Given  a  =  77.99,  i»  =  83.39,  C=  72*15'; 

fiiid.4=5ri5',  ^=56*30',         «?=95.24. 

2.   Given  ft  =  872.5,  r= 632.7,  ^=80**; 

find  ^ = 60**  45',  C= 39°  15',        a = 984.83. 

_    3.    Given  (7  =  17,  ft  =  12,  C=59°17'; 

find  .4  =77°  12' 53",    ^  =  43"  30' 7",    r=  14.987. 

4.  Given  ft  =  VB,  «?=  V3,  ^ =35°  53'; 

find  ^ = 93°  28'  36",    C=  50°  38'  24",  a  =  1.313. 

5.  Given  a  =0.917,  ft  =  0.312,  C=33°7'9"; 

find  .4  =132°  18' 27",  ^=14°  34' 24",  c=0.6775. 

6.  Given  CI  =  13.715,  <?  =  11.214,        ^  =  15°  22' 36"; 

find  .1  =  118°  55' 49",  C=  45°  41' 35",  ft =4.1554. 

7.  Given  ft =3000.9,  <r=1587.2,        ^=86°  4' 4"; 

find  ^=65°  13' 51",    (7=28°  42' 5",    a =3297.2. 

8.  Given  a  =4527,  ft=3465,  C=66°6'27"; 

find  J  =68°  29' 15",    ^ =45°  24' 18",  c= 4449. 

9.  Given  «  =  55.14,  ft=33.09,  C=30°24'; 

find  .1=117°  24' 32",  ^=32°  11' 28",  r=31.431. 

10.  Given  «=  47,99,  ft =33,14,        C=175°19'10"; 

find  J  =  2°46'8",        ^=1°54'42",  r=81.066. 

11.  If  two  sides  of  a  triangle  are  each  eqnal  to  6,  and  the 
included  angle  is  60°,  find  the  third  side. 

12.  If  two  sides  of  a  triangle  are  each  equal  to  6,  and  the 
included  angle  is  120"*,  find  the  third  side. 

13.  Apply  Solution  I.  to  the  case  in  which  a  is  equal  to  ft ; 
that  is,  the  case  in  which  the  triangle  is  isosceles. 

14.  If  two  sides  of  a  triangle  are  10  and  11,  and  the  included 
angle  is  50°,  find  the  third  side. 

15.  If  two  sides  of  a  triangle  are  43.301  and  25,  and  the 
included  angle  is  30°,  find  the  third  side. 

16.  In  order  to  find  the  distance  between  two  objects  A 
and  J^  separated  by  a  swamp,  a  station  C  was  chosen^  and  the 


74  TRIGONOMETRY. 

distances  C^  =  3825  yards,  CJ5  =  3475.6  yards,  together  with 
the  angle  ACB  =  62°  31',  were  measured.  Find  the  distance 
from  A  to  B. 

17.  Two  inaccessible  objects  A  and  B  are  each  viewed  from 
two  stations  C  and  D  on  the  same  side  of  AB  and  562  yards 
apart.  The  angle  ACB  is  62°  12',  BCD  41°  8',  ABB  60°  49', 
and  ADC  34°  51';  required  the  distance  AB. 

18.  Two  trains  start  at  the  same  time  from  the  same  station, 
and  move  along  straight  tracks  that  form  an  angle  of  30°,  one 
train  at  the  rate  of  30  miles  an  hour,  the  other  at  the  rate  of 
40  miles  an  hour.  How  far  apart  are  the  trains  at  the  end  of 
half  an  hour? 

19.  In  a  parallelogram  given  the  two  diagonals  5  and  6, 
and  the  angle  that  they  form  49°  18'.     Find  the  sides. 

20.  In  a  triangle  one  angle  =  139°  54',  and  the  sides  forming 
the  angle  have  the  ratio  5  :  9.     Find  the  other  two  angles. 

§  40.    Case  IY. 

Given  the  three  sides  <x,  b,  c;  find  the  angles  A,  B,  C. 
The  angles  may  be  found  directly  from  the  formulas  estab- 
lished in  §  34.     Thus,  from  the  formula 

a'  =  b'-{-c^~2bccosA 

we  have  cos^= — —rz 

Zbc 

From  this  equation  formulas  adapted  to  logarithmic  work 
are  deduced  as  follows  :  • 

For  the  sake  of  brevity,  let  a -\-b-{-c  =  2 s-,  then  b-\-G~a 
=  2  (s  —  a),  a  —  b-\-c  =  2(s  —  b),  and  a-\-b~c  =  2  (s  —  c). 

Then  the  value  of  1  —  cos  A  is 

^      b'-\-c'-a^^2bc-b^-c'-{-a^_a^-(b~cy 
2bc  2bG  ~         2bG 

^(a-j-b~c)(a~b  +  c)_2(s-b)(s  —  c) 
2bc  ~  be  ' 


THE    OBLIQUE    TRIANGLE.  75 

and  the  value  of  1  +  cos  A  is 
-f- 


^      b^-^c^-a^^2bc-{-b^-\-G^  —  a^_(b  +  cy  —  a^ 


2  be  2bG  2bG 

_(b  +  c-{-a)(b-{-c  —  a)_2s(s  —  a) 

~  2bG  ~        be 

But  from  Formulas  [16]  and  [17],  §  30,  it  follows  that 

1  —  cos^  =  2sin^-J-J,  and  1 +  cos^  =  2cos'^^^. 

2(s—b)(s  —  c)  2  s  (s  — a) 

.'.  2suv^A  =  —^ f^ ^>  and  2gos^^A  =  — H ^> 

bG  ^  be 

whence  smiA  =  yj^^~^^^^^~''\  [28] 

C08iA  =  V^-^^'  [29] 

and  by  [2]  tan  i  A  =  V^'t^Z^'  C^^] 

By  merely  changing  the  letters,- 

sin 


1  -r.  (s  —  ^)  (^  —  ^)       •     1  ^         l(s  —  <^)  (s  —  b) 

^  ^^  5(S— ^)  ^  ^^  5(S— C) 


There  is  then  a  choice  of  three  different  formulas  for  finding 
the  value  of  each  angle.  If  half  the  angle  is  very  near  0°, 
the  formula  for  the  cosine  will  not  give  a  very  accurate  result, 
because  the  cosines  of  angles  near  0°  differ  little  in  value  ;  and 
the  same  holds  true  of  the  formula  for  the  sine  when  half 
the  angle  is  very  near  90°.  Hence,  in  the  first  case  the 
formula  for  the  sine,  in  the  second  that  for  the  cosine,  should 
be  used. 

But,  in  general,  the  formulas  for  the  tangent  are  to  be 
preferred. 


76 


TRIGONOMETRY. 


It  is  not  necessary  to  compute  by  the  formulas  more  than 
two  angles  ;  for  the  third  may  then  be  found  from  the  equation 

There  is  this  advantage,  however,  in  computing  all  three 
angles  by  the  formulas,  that  we  may  then  use  the  sum  of  the 
angles  as  a  test  of  the  accuracy  of  the  results. 

In  case  it  is  desired  to  compute  all  the  angles,  the  formulas 
for  the  tangent  may  be  put  in  a  more  convenient  form. 

The  value  of  tan^^  may  be  written 


l(s  —  a)(s  —  b 
^  s  (s-ay 

Hence,  if  we  put 


!')(^-o) 


or 


1         (s-a)(s-l,)(s 
—  a^  s 


1 


V 


(s— a)  (s  —  b)  (s  —  c) 


r, 


we  have 


Likewise, 


tan^A: 

tan  ^  B  ■■ 


r 


[31] 
[32] 


tan^  C  = 


Examples. 
Given  c^  =  3.41,  h  =  2m,  c 


o\ 


1.58  :  find  the  ansrles. 

as'  ^ 


Using  Formula  [30],  and  the  corresponding  formula  for  tan  ^JB,  we 
may  arrange  the  work  as  follows : 


a  =  3.41 
6=2.60 
c=  1.58 
2s  =  7.59 
s=  3.795 
s-a=  0.385 
s-b=  1.195 
s  —  c  =  2.215 


cologs  =  9.42079 

colog  {s—  a)  =  0.41454 

log  (s  -  6)  =  0.07737 

log  (s  -  c)  =  0.34537 

2)0.25807 

log  taniJ.  =  0.12903 

iA=    53°  23' 20" 

A  =  106°  46'  40'' 

.  J.  4- 5  =153°  39' 64",  and 


colog  s=    9.42079—10 
log(s-a)=    9.58546-10 
colog(s-6)=    9.92263-10 
log  (s  -  c)  =    0.34537 

2)19.27425-20 
log  tan  i  I?  =    9.63713-10 


B  = 

C  =  26°  20'  6". 


23= 
46° 


26'  37' 
53'  14' 


THE    OBLIQUE    TRIANGLE. 


77 


2.    Solve  Example  1  by  finding  all  three  angles  by  the  use 
of  Formulas  [31]  and  [32]. 

Here  the  work  may  be  compactly  arranged  as  follows, 
log  tan  ^^,  etc.,  by  subtracting  log(s  — a),  etc.,  from  log 
adding  the  cologarithm  : 


a  =  3.41 

b  =  2.60 

c=  1.58 

2s  =7.59 


s  =  3.795 
s-a  =  0.385 
5-6=1.195 
5—  c  =  2.215 


log  (s- a)  =  9.58546 

log  (s- 6)  =  0.07737 

log  {s-  c)  =  0.34537 

colog  s  =  9.42079 

logr2  =  9.42899 

logr  =9.71450 


2s  =7.590  (proof). 


log  tan  iA  = 
log  tan  i  5  = 
log  tan  i  C  = 
iA  = 
iB  = 
iC  = 
A  = 
B  = 
C  = 
Proof,  A  +  B+  C  = 


,  if  we  find 
r  instead  of 

10.12903 
9.63713 
9.30912 
53°  23'  20'' 
23°  26'  37" 
13°  10'   3" 

106°  46'  40" 
46°  53'  14" 
26°  20'   6" 

180°   0'   0" 


Note.  Even  if  no  mistakes  are  n^ade  in  the  work,  the  sum  of  the 
three  angles  found  as  above  may  differ  very  slightly  from  180°  in  conse- 
quence of  the  fact  that  logarithmic  computation  is  at  best  only  a  method 
of  close  approximation.  When  a  difference  of  this  kind  exists  it  should 
be  divided  among  the  angles  according  to  the  probable  amount  of  error 
for  each  angle. 

Exercise  XX. 

Solve  the  following  triangles,  taking  the  three  sides  as  the 
given  parts  : 


1 

a 

6 

c 

yi 

B 

C 

51 

65 

20 

38°  52'  48" 

126°  52' 12" 

14°  15' 

2 

78 

101 

29 

32°  10'  54" 

136°  23'  50" 

11°"25'  16" 

3 

111 

145 

40 

27°  20' 32" 

143°  7' 48" 

9°  31'  40" 

4 

21 

26 
34 
50 

31 

49 
57 

42°  6'  13" 
16°  25'  36" 
46°  49'  35" 

56°  6' 36" 
30°  24' 
57°  59'  44" 

81°  47' 11" 

133° 10' 24" 

75° 10' 41" 

5 
6 

19 
43 

7 

37 

58 

79 

26°  0'29" 

43°  25'  20" 

110°  34' 11" 

8 

73 

82 

91 

49°  34'  58" 

58°  46'  58" 

71°  38'  4" 

9 

14.493 

55.4363 

66.9129  I 

8°  20' 

33°  40' 

138° 

10 

V5 

V6 

V7  ! 

51°  53'  12" 

59°  31'  48" 

68°  35' 

78  TRIGONOMETRY. 

11.  Given  a  =  6,  b  =  S,  c  =  10;  find  the  angles. 

12.  Given  a  =  6,  b  =  6,  c  =  10 ;  find  the  angles. 

13.  Given  a  =  6,  b  =  6,  c  =  6;     find  the  angles. 

14.  Given  a  =  6,  b  =  5,  c  =  12 ;  find  the  angles. 

15.  Given  a  =  2,  b=  VC,  c  =  VS  —  1 ;  find  the  angles. 

16.  Given  a  =  2,  b=  V6,  c  =  V3  + 1 ;  find  the  angles. 

17.  The  distances  between  three  cities  A,  B,  and  C  are  as 
follows  :  ^^  =  165  miles,  ^C=  72  miles,  and  BC^=  185  miles. 
B  is  due  east  from  A.  In  what  direction  is  C  from  A  ?  What 
two  answers  are  admissible  ? 

18.  Under  what  visual  angle  is  an  object  7  feet  long  seen 
by  an  observer  whose  eye  is  5  feet  from  one  end  of  the  object 
and  8  feet  from  the  other  end? 

19.  When  Formula  [28]  is  used  for  finding  the  value  of  an 
angle,  why  does  the  ambiguity  that  occurs  in  Case  II.  not  exist  ? 

20.  If  the  sides  of  a  triangle  are  3,  4,  and  6,  find  the  sine 
of  the  largest  angle. 

21.  Of  three  towns  A^  B,  and  C,  A  is  200  miles  from  B 
and  184  miles  from  C,  B  is  150  miles  due  north  from  C ;  how 
far  is  A  north  of  C? 

§  41.    Area  of  a  Triangle. 
Case  I.    When  two  sides  and  the  i7icluded  angle  are  given: 
In  the  triangle  ABC  (Fig.  31  or  32),  the  area 

F=icXCB. 
By  §11,  CD  =  a  sin  B. 

Therefore,  F  =  i  ac  sin  B.  [33] 

Also,  F=  ^ab  sin  C     and     F=^  bo  sin  A. 

Case  II.    When  a  side  and  the  two  adjacent  angles  are  given: 
By  §  33,  sinA:s\nC::a\c. 

a  sin  C 


Therefore,  c  = 


sin^ 


THE    OBLIQUE    TRIANGLE.  79 

Putting  this  value  of  c  in  Formula  [33],  we  have 
a^sinBsinC 

2sin(B  +  C)'  ^^ 

Case  III.    When  the  three  sides  of  a  triangle  are  given: 
By  §  29,  sini?  =  2sini^Xcosi-^. 

By  substituting  for  sin  ^  B  and  cos  |-  B  their  values  in  terms 
of  the  sides  given  in  §  40^  ,*: 

2     /- 

sin  i>  =  —  V^  (s  —  a)(s  —  b)  {s  —  c). 

By  putting  this  value  of  sin  B  in  [33],  we  have 


F=Vs(s-a)(s  — b)(s  — c).  [35] 

Case  IY.  When  the  three  sides  and  the  radius  of  the  circum- 
scribed circle,  or  the  radius  of  the  inscribed  circle,  are  given : 

If  R  denotes  the  radius  of  the  circumscribed  circle,  we  have, 
from  §  33, 

By  putting  this  value  of  sin  B  in  [33],  we  have 

F  =  ^-  [36] 

If  r  denotes  the  radius  of  the  inscribed  circle, 
divide   the   triangle   into  three   triangles   by  lines   from  the 
centre   of  this  circle  to  the  vertices;   then  the  altitude  of 
each  of  the  three  triangles  is  equal  to  r.     Therefore, 

F  =  ir(a  +  b  +  c)=js.  [37] 

By  putting  in  this  formula  the  value  of  F  given  in  [35], 


=4 


(s  —  a)  (s  —  b)  (s  —  c) 


whence  r,  in  [31]  §  40,  is  equal  to  the  radius  of  the  inscribed 
circle. 


80  TRIGONOMETRY. 

Exercise  XXI. 

Find  the  area : 

^\.  Given  a  =  4474.5,  ^>  =  2164.5,  (7=116°  30' 20". 

2.  Given  Z»  =  21.66,  c  =  36.94,  ^  =  66°  4' 19". 

3.  Given  a  =  510,  c  =  173,  ^  =  162°  30' 28". 

4.  Given  a  =  408,  ^^  =  41,  c  =  401. 

5.  Given  a  =  40,  ^^  =  13,  c  =  37. 
_^^6.  Given  a  =  624,  ^^  =  205,  c  =  445. 

^^1.  Given  &  =  149,  ^  =  70°  42' 30",  ^  =  39°  18' 28". 

8.  Given  61  =  215.9,  c  =  307.7,    ^1  =  25°  9' 31". 

9.  Given  ^^  =  8,  c  =  5,  ^  =  60°. 
10.  Given  a  =  7,  c  =  3,  ^  =  60°. 

^  11.    Given  a  =  60,    i?  =  40°  35' 12",    area  =  12  ;    find    the 
radius  of  the  inscribed  circle. 

12.  Obtain  a  formula  for  the  area  of  a  parallelogram  in 
terms  of  two  adjacent  sides  and  the  included  angle. 

13.  Obtain  a  formula  for  the  area  of  an  isosceles  trapezoid 
in  terms  of  the  two  parallel  sides  and  an  acute  angle. 

14.  Two  sides  and  included  angle  of  a  triangle  are  2416, 
1712,  and  30°;  and  two  sides  and  included  angle  of  another 
triangle  are  1948,  2848,  and  150°;  find  the  sum  of  their  areas. 

15.  The  base  of  an  isosceles  triangle  is  20,  and  its  area  is 
100  -^  V3  ;  find  its  angles. 

16.  Show  that  the  area  of  a  quadrilateral  is  equal  to  one 
half  the  product  of  its  diagonals  into  the  sine  of  their  included 
angle. 

Exercise  XXII. 

1.  From  a  ship  sailing  down  the  English  Channel  the  Eddy- 
stone  was  observed  to  bear  N.  33°  45'  W. ;  and  after  the  ship 
had  sailed  18  miles  S.  67°  30'  W.  it  bore  K  11°  15'  E.  Find 
its  distance  from  each  position  of  the  ship. 


THE    OBLIQUE    TRIANGLE.  81 

2.  Two  objects,  A  and  B,  were  observed  from  a  ship  to  be 
at  the  same  instant  in  a  line  bearing  N.  15°  E.  The  ship  then 
sailed  north-west  5  miles,  when  it  was  found  that  A  bore  due 
east  and  B  bore  north-east.     Find  the  distance  from  A  to  B. 

3.  A  castle  and  a  monument  stand  on  the  same  horizontal 
plane.  The  angles  of  depression  of  the  top  and  the  bottom  of 
the  monument  viewed  from  the  top  of  the  castle  are  40°  and 
80°;  the  height  of  the  castle  is  140  feet.  Find  the  height  of 
the  monument. 

4.  If  the  sun's  altitude  is  60°,  what  angle  must  a  stick 
make  with  the  horizon  in  order  that  its  shadow  in  a  horizontal 
plane  may  be  the  longest  possible? 

5.  If  the  sun's  altitude  is  30°,  find  the  length  of  the  longest 
shadow  cast  on  a  horizontal  plane  by  a  stick  10  feet  in  length. 

6.  In  a  circle  with  the  radius  3  find  the  area  of  the  part 
comprised  between  parallel  chords  whose  lengths  are  4  and  5. 
(Two  solutions.) 

7.  A  and  B,  two  inaccessible  objects  in  the  same  horizontal 
plane,  are  observed  from  a  balloon  at  (7,  and  from  a  point  D 
directly  under  the  balloon  and  in  the  same  horizontal  plane 
with  A  and  B.  If  CI)  =  2000  yards,  Z^ CD  =  10°  15' 10", 
Z  BCD  =  6°  7'  20",  Z  ADB  =  49°  34'  50",  find  AB. 

8.  A  and  B  are  two  objects  whose  distance,  on  account  of 
intervening  obstacles,  cannot  be  directly  measured.  At  the 
summit  C  of  a  hill,  whose  height  above  the  common  horizontal 
plane  of  the  objects  is  known  to  be  517.3  yards,  Z.ACB  is 
found  to  be  15°  13'  15".  The  angles  of  elevation  of  C  viewed 
from  A  and  B  are  21°  9'  18"  and  23°  15'  34"  respectively.  Find 
the  distance  from  A  to  B, 


CHAPTER   V. 

MISCELLANEOUS    EXAMPLES. 

Problems  in  Plane  Trigonometry. 

1.  The  angular  distance  of  any  object  from  a  horizontal 
plane,  as  observed  at  any  point  of  that  plane,  is  the  angle 
which  a  line  drawn  from  the  object  to  the  point  of  observa- 
tion makes  with  the  plane.  If  the  object  observed  is  situated 
above  the  horizontal  plane  (that  is,  if  it  is  farther  from  the 
earth's  centre  than  the  plane  is),  its  angular  distance  from 
the  plane  is  called  its  ayigle  of  elevation.  If  the  object  is 
below  the  plane,  its  angular  distance  from  the  plane  is  called 
its  angle  of  depression.  These  angles  are  evidently  vertical 
angles. 

If  two  objects  are  in  the  same  horizontal  plane  with  the 
point  of  observation,  the  angular  distance  of  one  object  from 
the  other  is  called  its  hearing  from  that  object. 

If  two  objects  are  not  in  the  same  horizontal  plane  with 
either  each  other  or  the  point  of  observation,  we  may  suppose 
vertical  lines  to  be  passed  through  the  two  objects,  and  to 
meet  the  horizontal  plane  of  the  point  of  observation  in  two 
points.  The  angular  distance  of  these  two  points  is  the 
bearing  of  either  of  the  objects  from  the  other.  It  may 
also  be  called  the  horizontal  distance  of  one  object  from  the 
other. 

Note.  "Problems  in  Plane  Trigonometry  "  are  selected  from  those 
published  by  Mr.  Charles  W.  Sever,  Cambridge,  Mass.  The  full  set  can 
be  obtained  from  him  in  pamphlet  form. 


MISCELLANEOUS    EXAMPLES.  83 


Eight  Triangles. 

2.  The  angle  of  elevation  of  a  tower  is  48°  19'  14",  and  the 
distance  of  its  base  from  the  point  of  observation  is  95  ft. 
Find  the  height  of  the  tower,  and  the  distance  of  its  top  from 
the  point  of  observation. 

3.  From  a  mountain  1000  ft.  high,  the  angle  of  depression 
of  a  ship  is  77°  35'  11".  Find  the  distance  of  the  ship  from 
the  summit  of  the  mountain. 

4.  A  flag-staff  90  ft.  high,  on  a  horizontal  plane,  casts  a 
shadow  of  117  ft.     Find  the  altitude  of  the  sun. 

5.  When  the  moon  is  setting  at  any  place,  the  angle  at  the 
moon  subtended  by  the  earth's  radius  passing  through  that 
place  is  57'  3".  If  the  earth's  radius  is  3956.2  miles,  what  is 
the  moon's  distance  from  the  earth's  centre  ? 

6.  The  angle  at  the  earth's  centre  subtended  by  the  sun's 
radius  is  IG'  2",  and  the  sun's  distance  is  92,400,000  miles. 
Find  the  sun's  diameter  in  miles. 

7.  The  latitude  of  Cambridge,  Mass.,  is  42°  22'  49".  What 
is  the  length  of  the  radius  of  that  parallel  of  latitude  ? 

8.  At  what  latitude  is  the  circumference  of  the  parallel  of 
latitude  half  of  that  of  the  equator  ? 

9.  In  a  circle  with  a  radius  of  6.7  is  inscribed  a  regular 
polygon  of  thirteen  sides.     Find  the  length  of  one  of  its  sides. 

10.  A  regular  heptagon,  one  side  of  which  is  5.73,  is 
inscribed  in  a  circle.     Find  the  radius  of  the  circle. 

11.  A  tower  93.97  ft.  high  is  situated  on  the  bank  of  a 
river.  The  angle  of  depression  of  an  object  on  the  opposite 
bank  is  25°  12'  54".     Find  the  breadth  of  the  river. 


84  TRIGONOMETRY. 

12.  From  a  tower  58  ft.  high  the  angles  of  depression  of 
two  objects  situated  in  the  same  horizontal  line  with  the  base 
of  the  tower,  and  on  the  same  side,  are  30°  13'  18"  and  45° 
46'  14".     Find  the  distance  between  these  two  objects. 

13.  Standing  directly  in  front  of  one  corner  of  a  flat-roofed 

house,  which  is  150  ft.  in  length,  I  observe  that  the  horizontal 

angle  which  the  length  subtends  has  for  its  cosine  V^,  and 

that  the  vertical  angle  subtended  by  its  height  has  for  its  sine 
3 

-—='     What  is  the  height  of  the  house  ? 
V34 

14.  A  regular  pyramid,  with  a  square  base,  has  a  lateral  edge 
150  ft.  in  length,  and  the  length  of  a  side  of  its  base  is  200  ft. 
Find  the  inclination  of  the  face  of  the  pyramid  to  the  base. 

15.  From  one  edge  of  a  ditch  36  ft.  wide,  the  angle  of 
elevation  of  a  wall  on  the  opposite  edge  is  62°  39'  10".  Find 
the  length  of  a  ladder  which  will  reach  from  the  point  of 
observation  to  the  top  of  the  wall. 

16.  The  top  of  a  flag-staff  has  been  broken  off,  and  touches 
the  ground  at  a  distance  of  15  ft.  from  the  foot  of  the  staff. 
The  length  of  the  broken  part  being  39  ft.,  find  the  whole 
lehgth  of  the  staff'. 

17.  From  a  balloon,  which  is  directly  above  one  town,  is 
observed  the  angle  of  depression  of  another  town,  10°  14'  9". 
The  towns  being  8  miles  apart,  find  the  height  of  the  balloon. 

18.  From  the  top  of  a  mountain  3  miles  high  the  angle  of 
depression  of  the  most  distant  object  which  is  visible  on  the 
earth's  surface  is  found  to  be  2°  13'  50".  Find  the  diameter 
of  the  earth. 

19.  A  ladder  40  ft.  long  reaches  a  window  33  ft.  high,  on 
one  side  of  a  street.  Being  turned  over  upon  its  foot,  it 
reaches  another  window  21  ft.  high,  on  the  opposite  side  of 
the  street.     Find  the  width  of  the  street. 


MISCELLANEOUS    EXAMPLES.  85 

20.  The  height  of  a  house  subtends  a  right  angle  at  a 
window  on  the  other  side  of  the  street ;  and  the  elevation  of 
the  top  of  the  house,  from  the  same  point,  is  60°.  The  street 
is  30  ft.  wide.     How  high  is  the  house  ? 

21.  A  lighthouse  54  ft.  high  is  situated  on  a  rock.  The 
elevation  of  the  top  of  the  lighthouse,  as  observed  from  a  ship, 
is  4°  52',  and  the  elevation  of  the  top  of  the  rock  is  4°  2'. 
Find  the  height  of  the  rock,  and  its  distance  from  the  ship. 

22.  A  man  in  a  balloon  observes  the  angle  of  depression  of 
an  object  on  the  ground,  bearing  south,  to  be  35°  30';  the 
balloon  drifts  2J  miles  east  at  the  same  height,  when  the  angle 
of  depression  of  the  same  object  is  23°  14'.  Find  the  height 
of  the  balloon. 

23.  A  man  standing  south  of  a  tower,  on  the  same  horizon- 
tal plane,  observes  its  elevation  to  be  54°  16' ;  he  goes  east 
100  yds.,  and  then  finds  its  elevation  is  50°  8'.  Find  the 
height  of  the  tower. 

24.  The  elevation  of  a  tower  at  a  place  A  south  of  it  is 
30°;  and  at  a  place  B,  west  of  A,  and  at  a  distance  of  a  from 
it,  the  elevation  is  18°.     Show  that  the  height  of  the  tower  is 

the  tangent  of  18°  being 


V(2  +  2V5)  V(10+2V5) 

25.    A  pole  is  fixed  on  the  top  of  a  mound,  and  the  angles 

of  elevation  of  the  top  and  the  bottom  of  the  pole  are  60°  and 

30°  respectively.     Prove  that  the  length  of  the  pole  is  twice 

the  height  of  the  mound. 

2Q.    At  a  distance  (a)  from  the  foot  of  a  tower,  the  angle 

of  elevation  {A)  of  the  top  of  the  tower  is  the  complement  of 

the  angle  of  elevation  of  a  flag-staff  on  top  of  it.     Show  that 

the  length  of  the  staff  is  2  a  cot  2  A. 

27.    A  line  of  true  level  is  a  line  every  point  of  which  is 
equally  distant  from  the  centre  of  the  earth.     A  line  drawn 


86  TRIGONOMETRY. 

tangent  to  a  line  of  true  level  at  any  point  is  a  line  of 
apparent  level.  If  at  any  point  both  these  lines  are  drawn, 
and  extended  one  mile,  find  the  distance  they  are  then  apart. 

28.  In  Problem  2,  determine  the  effect  upon  the  computed 
height  of  the  tower,  of  an  error  in  either  the  angle  of  elevation 
or  the  measured  distance. 

Oblique  Triangles. 

29.  To  determine  the  height  of  an  inaccessible  object 
situated  on  a  horizontal  plane,  by  observing  its  angles  of 
elevation  at  two  points  in  the  same  line  with  its  base,  and 
measuring  the  distance  between  these  two  points. 

X  30.  The  angle  of  elevation  of  an  inaccessible  tower,  situated 
on  a  horizontal  plane,  is  63°  26';  at  a  point  500  ft.  farther 
from  the  base  of  the  tower  the  elevation  of  its  top  is  32"^  14'. 
Find  the  height  of  the  tower. 

31.  A  tower  is  situated  on  the  bank  of  a  river.  From  the 
opposite  bank  the  angle  of  elevation  of  the  tower  is  60°  13', 
and  from  a  point  40  ft.  more  distant  the  elevation  is  50°  19'. 
Find  the  breadth  of  the  river. 

_  32.  A  ship  sailing  north  sees  two  lighthouses  8  miles  apart, 
in  a  line  due  west ;  after  an  hour's  sailing,  one  lighthouse 
bears  S.W.,  and  the  other  S.S.W.     Find  the  ship's  rate. 

33.  To  determine  the  height  of  an  accessible  object  situated 
on  an  inclined  plane. 

34.  At  a  distance  of  40  ft.  from  the  foot  of  a  tower  on  an 
inclined  plane,  the  tower  subtends  an  angle  of  41°  19';  at  a 
point  60  ft.  farther  away,  the  angle  subtended  by  the  tower 
is  23°  45'.     Find  the  height  of  the  tower. 

35.  A  tower  makes  an  angle  of  113°  12'  with  the  inclined 
plane  on  which  it  stands ;  and  at  a  distance  of  89  ft.  from  its 
base,  measured  down  the  plane,  the  angle  subtended  by  the 
tower  is  23°  27'.     Find  the  height  of  the  tower. 


MISCELLANEOUS    EXAMPLES.  87 

36.  From  the  top  of  a  house  42  ft.  high,  the  angle  of 
elevation  of  the  top  of  a  pole  is  14°  13';  at  the  bottom  of  the 
house  it  is  23°  19'.     Find  the  height  of  the  pole. 

37.  The  sides  of  a  triangle  are  17,  21,  28 ;  prove  that  the 
length  of  a  line  bisecting  the  greatest  side  and  drawn  from 
the  opposite  angle  is  13. 

38.  A  privateer,  10  miles  S.W.  of  a  harbor,  sees  a  ship  sail 
from  it  in  a  direction  S.  80°  E.,  at  a  rate  of  9  miles  an  hour. 
In  what  direction,  and  at  what  rate,  must  the  privateer  sail 
in  order  to  come  up  with  the  ship  in  1^  hours? 

39.  A  person  goes  70  yds.  up  a  slope  of  1  in  3J  from  the 
edge  of  a  river,  and  observes  the  angle  of  depression  of  an 
object  on  the  opposite  bank  to  be  2^°.  Find  the  breadth  of 
the  river. 

40.  The  length  of  a  lake  subtends,  at  a  certain  point,  an 
angle  of  46°  24',  and  the  distances  from  this  point  to  the  two 
extremities  of  the  lake  are  346  and  290  ft.  Find  the  length 
of  the  lake. 

41.  Two  ships  are  a  mile  apart.  The  angular  distance  of 
the  first  ship  from  a  fort  on  shore,  as  observed  from  the  second 
ship,  is  35°  14'  10";  the  angular  distance  of  the  second  ship 
from  the  fort,  observed  from  the  first  ship,  is  42°  11'  53". 
Find  the  distance  in  feet  from  each  ship  to  the  fort. 

42.  Along  the  bank  of  a  river  is  drawn  a  base  line  of  500 
feet.  The  angular  distance  of  one  end  of  this  line  from  an 
object  on  the  opposite  side  of  the  river,  as  observed  from  the 
other  end  of  the  line,  is  53°;  that  of  the  second  extremity 
from  the  same  object,  observed  at  the  first,  is  79°  12'.  Find 
the  perpendicular  breadth  of  the  river. 

43.  A  vertical  tower  stands  on  a  declivity  inclined  15°  to 
the  horizon.  A  man  ascends  the  declivity  80  ft.  from  the  base 
of  the  tower,  and  finds  the  angle  then  subtended  by  the  tower 
to  be  30°.     Find  the  height  of  the  tower. 


88  TRIGONOMETRY. 

44.  The  angle  subtended  by  a  tower  on  an  inclined  plane  is, 
at  a  certain  point,  42°  17';  325  ft.  farther  down,  it  is  21°  47'. 
The  inclination  of  the  plane  is  8°  53'.  Find  the  height  of  the 
tower. 

45.  A  cape  bears  north  by  east,  as  seen  from  a  ship.  The 
ship  sails  northwest  30  miles,  and  then  the  cape  bears  east. 
How  far  is  it  from  the  second  point  of  observation  ? 

46.  Two  observers,  stationed  on  opposite  sides  of  a  cloud, 
observe  its  angles  of  elevation  to  be  44°  ^^^  and  36°  4'.  Their 
distance  from  each  other  is  700  ft.  What  is  the  linear  height 
of  the  cloud  ? 

47.  From  a  point  B  at  the  foot  of  a  mountain,  the  elevation 
of  the  top  A  is  60°.  After  ascending  the  mountain  one  mile, 
at  an  inclination  of  30°  to  the  horizon,  and  reaching  a  point  (7, 
the  angle  ACB  is  found  to  be  135°.  Find  the  height  of  the 
mountain  in  feet. 

48.  From  a  ship  two  rocks  are  seen  in  the  same  right  line 
with  the  ship,  bearing  IST.  15°  E.  After  the  ship  has  sailed 
northwest  5  miles,  the  first  rock  bears  east,  and  the  second 
northeast.     Find  the  distance  between  the  rocks. 

49.  From  a  window  on  a  level  with  the  bottom  of  a  steeple 
the  elevation  of  the  steeple  is  40°,  and  from  a  second  window 
18  ft.  higher  the  elevation  is  37°  30'.  Find  the  height  of  the 
steeple. 

50.  To  determine  the  distance  between  two  inaccessible 
objects  by  observing  angles  at  the  extremities  of  a  line  of 
known  length. 

51.  Wishing  to  determine  the  distance  between  a  church  A 
and  a  tower  B,  on  the  opposite  side  of  a  river,  I  measure  a 
line  CD  along  the  river  ((7  being  nearly  opposite  A),  and 
observe  the  angles  ACB,  58°  20';  ACD,  95°  20';  ADB,  53°  30'; 
BDC,  98°  45'.     CD  is  600  ft.    What  is  the  distance  required  ? 


MISCELI.ANEOUS    EXAMPLES.  89 

52.  Wishing  to  find  the  height  of  a  summit  A,  I  measure  a 
horizontal  base  line  CD,  440  yds.  At  C,  the  elevation  of  A  is 
37°  18',  and  the  horizontal  angle  between  D  and  the  summit  is 
76°  18';  at  D,  the  horizontal  angle  between  C  and  the  summit 
is  67°  14'.     Find  the  height. 

53.  A  balloon  is  observed  from  two  stations  3000  ft.  apart. 
At  the  first  station  the  horizontal  angle  of  the  balloon  and  the 
other  station  is  75°  25',  and  the  elevation  of  the  balloon  is  18°. 
The  horizontal  angle  of  the  first  station  and  the  balloon, 
measured  at  the  second  station,  is  64°  30'.  Find  the  height 
of  the  balloon. 

54.  Two  forces,  one  of  410  pounds,  and  the  other  of  320 
pounds,  make  an  angle  of  51°  37'.  Find  the  intensity  and  the 
direction  of  their  resultant. 

55.  An  unknown  force,  combined  with  one  of  128  pounds, 
produces  a  resultant  of  200  pounds,  and  this  resultant  makes 
an  angle  of  18°  24'  with  the  known  force.  Find  the  intensity 
and  direction  of  the  unknown  force. 

56.  At  two  stations,  the  height  of  a  kite  subtends  the  same 
angle  A.  The  angle  which  the  line  joining  one  station  and 
the  kite  subtends  at  the  other  station  is  B;  and  the  distance 
between  the  two  stations  is  a.  Show  that  the  height  of  the 
kite  is  ^a  sin  A  sec  B. 

57.  Two  towers  on  a  horizontal  plane  are  120  ft.  apart.  A 
person  standing  successively  at  their  bases  observes  that  the 
angular  elevation  of  one  is  double  that  of  the  other  ;  but,  when 
he  is  half-way  between  them,  the  elevations  are  complementary. 
Prove  that  the  heights  of  the  towers  are  90  and  40  ft. 

5S.  To  find  the  distance  of  an  inaccessible  point  C  from 
either  of  two  points  A  and  B,  having  no  instruments  to 
measure  angles.  Prolong  CA  to  a,  and  CB  to  b,  and  join 
AB,  Ah,  and  Ba.  Measure  AB,  500  j  aA,  100;  aB,  560; 
hB,  100  ;  and  Ab,  550. 


90  TRIGONOMETRY. 

59.  Two  inaccessible  points  A  and  B,  are  visible  from  D, 
but  no  other  point  can  be  found  whence  both  are  visible. 
Take  some  point  C,  whence  A  and  D  can  be  seen,  and  meas- 
ure CD,  200  ft. ;  ADC,  89°  ;  ACD,  50°  30'.  Then  take  some 
point  JS,  whence  D  and  B  are  visible,  and  measure  DE,  200 ; 
BDE,  54°  30' ;  BBD,  88°  30'.  At  D  measure  ADD,  72°  30'. 
Compute  the  distance  AB. 

60.  To  compute  the  horizontal  distance  between  two  inac- 
cessible points  A  and  B,  when  no  point  can  be  found  whence 
both  can  be  seen.  Take  two  points  C  and  D,  distant  200  yds., 
so  that  A  can  be  seen  from  C,  and  B  from  D.  From  C  meas- 
ure CF,  200  yds.  to  F,  whence  A  can  be  seen ;  and  from  D 
measure  DF,  200  yds.  to  F,  whence  B  can  be  seen.  Measure 
AFC,  83°;  ACD,  53°  30';  ACF,  54°  31';  BDF,  54°  30'; 
BDC,  156°  25';  DFB,  88°  30'. 

61.  A  column  in  the  north  temperate  zone  is  east-southeast 
of  an  observer,  and  at  noon  the  extremity  of  its  shadow  is 
northeast  of  him.  The  shadow  is  80  ft.  in  length,  and  the 
elevation  of  the  column,  at  the  observer's  station,  is  45°. 
Find  the  height  of  the  column. 

62.  From  the  top  of  a  hill  the  angles  of  depression  of  two 
objects  situated  in  the  horizontal  plane  of  the  base  of  the  hill 
are  45°  and  30° ;  and  the  horizontal  angle  between  the  two 
objects  is  30°.  Show  that  the  height  of  the  hill  is  equal  to 
the  distance  between  the  objects. 

63.  Wishing  to  know  the  breadth  of  a  river  from  A  to  B, 
I  take  AC,  100  yds.  in  the  prolongation  of  BA,  and  then  take 
CD,  200  yds.  at  right  angles  to  AC.  The  angle  BDA  is  37° 
18'  30".     Find  AB. 

64.  The  sum  of  the  sides  of  a  triangle  is  100.  The  angle 
at  A  is  double  that  of  B,  and  the  angle  at  B  is  double  that  at 
C.     Determine  the  sides. 


MISCELLANEOUS   EXAMPLES.  91 

65.  If  sin^A  -f  5  cos^^  =  3,  find  A. 

66.  If  sinM  =  m  cos  A  —  n,  find  cos  A. 

67.  Given  sin  ^  =  m  sin  B,  and  tan  A  =  n  tan  ^,  find  sin  A 
and  cos  ^. 

68.  If  tanM  +  4  sin^^  =  6,  find  ^. 

69.  If  sin  ^  =  sin  2  A,  find  A. 

70.  If  tan  2  ^  =  3  tan  ^,  find  A. 

71.  Prove  that  tan  50°  +  cot  50°  =  2  sec  10°. 

72.  Given  a  regular  polygon  of  n  sides,  and  calling  one  of 
them  a,  find  expressions  for  the  radii  of  the  inscribed  and  the 
circumscribed  circles  in  terms  of  n  and  a. 

If  P,  H,  D  are  the  sides  of  a  regular  inscribed  pentagon, 
hexagon,  and  decagon,  prove  P^^H^-\-D\ 

Areas. 

73.  Obtain  the  formula  for  the  area  of  a  triangle,  given 
two  sides  b,  c,  and  the  included  angle  A. 

74.  Obtain  the  formula  for  the  area  of  a  triangle,  given 
two  angles  A  and  B,  and  included  side  c. 

75.  Obtain  the  formula  for  the  area  of  a  triangle,  given 
the  three  sides. 

76.  If  a  is  the  side  of  an  equilateral  triangle,  show  that 

its  area  is  — - — 
4 

77.  Two  consecutive  sides  of  a  rectangle  are  52.25  ch.  and 
38.24  ch.     Find  its  area. 

78.  Two  sides  of  a  parallelogram  are  59.8  ch.  and  37.05  ch., 
and  the  included  angle  is  72°  10'.     Find  the  area. 

79.  Two  sides  of  a  parallelogram  are  15.36  ch.  and  11.46 
ch.,  and  the  included  angle  is  47°  30'.     Find  its  area. 


92  TRIGONOMETRY. 

80.  Two  sides  of  a  triangle  are  12.38  ch.  and  6.78  ch.,  and 
the  included  angle  is  46°  24'.     Find  the  area. 

81.  Two  sides  of  a  triangle  are  18.37  ch.  and  13.44  ch., 
and  they  form  a  right  angle.     Find  the  area. 

82.  Two  angles  of  a  triangle  are  76°  54'  and  57°  33'  12", 
and  the  included  side  is  9  ch.     Find  the  area. 

83.  Two  sides  of  a  triangle  are  19.74  ch.  and  17.34  ch. 
The  first  bears  N.  S2°  30'  W. ;  the  second  S.  24°  15'  E.  Find 
the  area. 

84.  The  three  sides  of  a  triangle  are  49  ch.,  50.25  ch.,  and 
25.69  ch.     Find  the  area. 

85.  The  three  sides  of  a  triangle  are  10.64  ch.,  12.28  ch., 
and  9  ch.     Find  the  area. 

86.  The  sides  of  a  triangular  field,  of  which  the  area  is 
14  acres,  are  in  the  ratio  of  3,  5,  7.     Find  the  sides. 

87.  In  the  quadrilateral  ABCD  we  have  AB,  17.22  ch. ; 
AD,  7.45  ch. ;  CD,  14.10  ch. ;  BC,  5.25  ch. ;  and  the  diagonal 
AC,  15.04  ch.     Find  the  area. 

88.  The  diagonals  of  a  quadrilateral  are  a  and  h,  and  they 
intersect  at  an  angle  D.  Show  that  the  area  of  the  quadri- 
lateral is  ^ah  sin  D. 

89.  The  diagonals  of  a  quadrilateral  are  34  and  56,  inter- 
secting at  an  angle  of  67°.     Find  the  area. 

90.  The  diagonals  of  a  quadrilateral  are  75  and  49,  inter- 
secting at  an  angle  of  42°.     Find  the  area. 

91.  Show  that  the  area  of  a  regular  polygon  of  n  sides,  of 

^.  ^  .         .    nw^     ,180° 

wnicn  one  is  a,  is  ~r-  cot 

4  n 

92.  One  side  of  a  regular  pentagon  is  25.     Find  the  area. 

93.  One  side  of  a  regular  hexagon  is  32.     Find  the  area. 


MISCELLANEOUS    EXAMPLES.  93 

94.  One  side  of  a  regular  decagon  is  46.     Eind  the  area. 

95.  Find  the  area  of  a  circle  whose  circumference  is  74  ft. 

96.  Find  the  area  of  a  circle  whose  radius  is  125  ft. 

97.  In  a  circle  with  a  diameter  of  125  ft.  find  the  area  of  a 
sector  with  an  arc  of  22°, 

98.  In  a  circle  with  a  radius  of  44  ft.  find  the  area  of  a 
sector  with  an  arc  of  25°. 

99.  In  a  circle  with  a  diameter  of  50  ft.  find  the  area  of  a 
segment  with  an  arc  of  280°. 

100.  Find  the  area  of  a  segment  (less  than  a  semicircle),  of 
which  the  chord  is  20,  and  the  distance  of  the  chord  from  the 
middle  point  of  the  smaller  arc  is  2. 

101.  If  r  is  the  radius  of  a  circle,  the  area  of  a  regular 
circumscribed  polygon  of  n  sides  is  nr^  tan 

The  area  of  a  regular  inscribed  polygon  is  -  r^  sin 

^1  ft 

102.  If  a  is  a  side  of  a  regular  polygon  of  7t.  sides,  the  area 
of  the  inscribed  circle  is  — —  cot^ 


4  n 


TTd 


The  area  of  the  circumscribed  circle  is  —r-  csc^ 


180^ 


4  n 

103.  The  area  of  a  regular  polygon  inscribed  in  a  circle  is 
to  that  of  the  circumscribed  polygon  of  the  same  number  of 
sides  as  3  to  4.     Find  the  number  of  sides. 

104.  The  area  of  a  regular  polygon  inscribed  in  a  circle  is 
a  geometric  mean  between  the  areas  of  an  inscribed  and  a 
circumscribed  regular  polygon  of  half  the  number  of  sides. 

105.  The  area  of  a  circumscribed  regular  polygon  is  an 
harmonic  mean  between  the  areas   of  an  inscribed  regular 


94  TRIGONOMETRY. 

polygon  of  the  same  number  of  sides,  and  of  a  circumscribed 
regular  polygon  of  half  that  number. 

106.  The  perimeter  of  a  circumscribed  regular  triangle  is 
double  that  of  the  inscribed  regular  triangle. 

107.  The  square  described  about  a  circle  is  four-thirds  the 
inscribed  dodecagon. 

108.  Two  sides  of  a  triangle  are  3  and  12,  and  the  included 
angle  is  30°.  Find  the  hypotenuse  of  an  isosceles  right  tri- 
angle of  equal  area. 

Plane  Sailing. 

109.  Plane  Sailing  is  that  branch  of  Navigation  in  which 
the  surface  of  the  earth  is  considered  a  plane.  The  problems 
which  arise  are  therefore  solved  by  the  methods  of  Plane 
Trigonometry. 

The  following  definitions  will  explain  the  technical  terms 
which  are  employed : 

The  difference  of  latitude  of  two  places  is  the  arc  of  a 
meridian  comprehended  between  the  parallels  of  latitude 
passing  through  those  places. 

The  departure  between  two  meridians  is  the  arc  of  a 
parallel  of  latitude  comprehended  between  those  meridians. 
It  evidently  diminishes  as  the  distance  from  the  equator  at 
which  it  is  measured  increases. 

When  a  ship  sails  in  such  a  manner  as  to  cross  successive 
meridians  at  the  same  angle,  it  is  said  to  sail  on  a  rhumb-line. 
The  constant  angle  which  this  line  makes  with  the  meridians 
is  called  the  course,  and  the  distance  between  two  places  is 
measured  on  a  rhumb-line. 

If  we  neglect  the  curvature  of  the  earth,  and  consider  the 
distance,  departure,  and  difference  of  latitude  of  two  places  to 


MISCELLANEOUS    EXAMPLES.  95 

be  straight  lines,  lying  in  one  plane,  they  will  form  a  right 
triangle,  called  the  triangle  of  plane  sailing.  If  ABD  be  a 
plane  triangle,  right-angled  at  Z>,  and  AD  represent  the  dif- 
ference of  latitude  of  A  and  B,  DAB  will  be  the  course  from 
A  to  B,  AB  the  distance,  and  DB  the  departure,  measured 
from  B,  between  the  meridian  of  A  and  that  of  B. 

110.  Taking  the  earth's  equatorial  diameter  to  be  7925.6 
miles,  find  the  length  in  feet  of  the  arc  of  one  minute  of  a 
great  circle.* 

111.  A  ship  sails  from  latitude  43°  45'  S.,  on  a  course  N. 
by  E.,  2345  miles.  Find  the  latitude  reached,  and  the 
departure  made. 

112.  A  ship  sails  from  latitude  1°  45'  N.,  on  a  course  S.E. 
by  E.,  and  reaches  latitude  2°  31'  S.  Find  the  distance,  and 
the  departure. 

113.  A  ship  sails  from  latitude  13°  17'  S.,  on  a  course  N.E. 
by  E.  f  E.,  until  the  departure  is  207  miles.  Find  the  distance, 
and  the  latitude  reached. 

114.  A  ship  sails  on  a  course  between  S.  and  E.,  244  miles, 
leaving  latitude  2°  52'  S.,  and  reaching  latitude  5°  8'  S.  Find 
the  course,  and  the  departure. 

115.  A  ship  sails  from  latitude  32°  18'  N.,  on  a  course 
between  N.  and  W.,  making  a  distance  of  344  miles,  and  a 
departure  of  103  miles.  Find  the  course,  and  the  latitude 
reached. 

116.  A  ship  sails  on  a  course  between  S.  and  E.,  making  a 
difference  of  latitude  136  miles,  and  a  departure  203  miles. 
Find  the  distance,  and  the  course. 

117.  A  ship  sails  due  north  15  statute  miles  an  hour,  for 
one  day.     What  is  the  distance,  in  a  straight  line,  from  the 

*  The  length  of  the  arc  of  one  minute  of  a  great  circle  of  the  earth  is 
called  a  geographical  mile,  or  a  knot.  In  the  following  problems,  this  is 
the  distance  meant  by  the  term  "mile,"  unless  otherwise  stated. 


96  TRIGONOMETRY. 

point  left  to  the  point  reached  ?     (Take  earth's  radins,  3962.8 
statute  miles.) 

Parallel  and  Middle  Latitude  Sailing. 

118.  The  difference  of  longitude  of  two  places  is  the  angle 
at  the  pole  made  by  the  meridians  of  these  two  places ;  or,  it 
is  the  arc  of  the  equator  comprehended  between  these  two 
meridians. 

119.  In  Parallel  Sailing,  a  vessel  is  supposed  to  sail  on  a 
parallel  of  latitude ;  that  is,  either  due  east  or  due  west.  The 
distance  sailed  is,  in  this  case,  evidently  the  departure  made ; 
and  the  difference  of  longitude  made  depends  on  the  solution 
of  the  following  problem  : 

120.  Given  the  departure  between  any  two  meridians  at 
any  latitude,  find  the  angle  which  those  meridians  make,  or 
the  difference  of  longitude  of  any  point  on  one  meridian  from 
any  point  on  the  other.  (The  earth  is  considered  to  be  a 
perfect  sphere,  and  the  solution  depends  on  simple  geometric 
and  trigonometric  principles.  Cf.  Problem  7.)  The  solution 
gives  the  following  formula : 

Diff.  long.  =  depart.  X  sec.  lat. 

121.  A  ship  in  latitude  42°  16'  N.,  longitude  72°  16'  W., 
sails  due  east  a  distance  of  149  miles.  What  is  the  position 
of  the  point  reached? 

122.  A  ship  in  latitude  44°  49'  S.,  longitude  119°  42'  E., 
sails  due  west  until  it  reaches  longitude  117°  16'  E.  Find  the 
distance  made. 

123.  In  Middle  Latitude  Sailing,  the  departure  between  two 
places,  not  on  the  same  parallel  of  latitude,  is  considered  to 
be,  approximately,  the  departure  between  the  meridians  of 
those  places,  measured  on  that  parallel  of  latitude  which  lies 
midway  between  the  parallels  of  the  two  places.     Except  in 


MISCELLANEOUS    EXAMPLES.  97 

very  high  latitudes  or  excessive  runs,  such  an  assumption 
produces  no  great  error.  By  the  formula  of  Example  120, 
then,  we  shall  have 

Diff.  long.  =  depart.  X  sec.  mid.  lat. 

124.  A  ship  leaves  latitude  31°  14'  JST.,  longitude  42°  19'  W., 
and  sails  E.N.E.  325  miles.     Find  the  position  reached. 

125.  Find  the  bearing  and  distance  of  Cape  Cod  from 
Havana.  (Cape  Cod,  42°  2'  N.,  70°  3'  W. ;  Havana,  23°  9'  N., 
82°  22'  W.) 

126.  Leaving  latitude  49°  57'  N.,  longitude  15°  16'  W.,  a 
ship  sails  between  S.  and  W.  till  the  departure  is  194  miles, 
and  the  latitude  is  47°  18'  N.  Find  the  course,  distance,  and 
longitude  reached. 

127.  Leaving  latitude  42°  30'  N.,  longitude  58°  51'  W.,  a 
ship  sails  S.E.  by  S.  300  miles.     Find  the  position  reached. 

128.  Leaving  latitude  49°  57'  N.,  longitude  30°  W.,  a  ship 
sails  S.  39°  W.,  and  reaches  latitude  47°  44'  N.  Find  the 
distance,  and  longitude  reached. 

129.  Leaving  latitude  37°  K,  longitude  32°  16' W.,  a  ship 
sails  between  N.  and  W.  300  miles,  and  reaches  latitude  41°  N. 
Find  the  course,  and  longitude  reached. 

130.  Leaving  latitude  50°  10'  S.,  longitude  30°  E.,  a  ship 
sails  E.S.E.,  making  160  miles'  departure.  Find  the  distance, 
and  position  reached. 

131.  Leaving  latitude  49°  30'  K,  longitude  25°  W.,  a  ship 
sails  between  S.  and  E.  215  miles,  making  a  departure  of  167 
miles.     Find  the  course,  and  position  reached. 

132.  Leaving  latitude  43°  S.,  longitude  21°  W.,  a  ship  sails 
273  miles,  and  reaches  latitude  40°  17'  S.  What  are  the  two 
courses  and  longitudes,  either  one  of  which  will  satisfy  the 
data? 


98  TRIGONOMETRY. 

133.  Leaving  latitude  17°  N.,  longitude  119°  E.,  a  ship  sails 
219  miles,  making  a  departure  of  162  miles.  What  four  sets 
of  answers  do  we  get  ? 

134.  A  ship  in  latitude  30°  sails  due  east  360  statute  miles. 
What  is  the  shortest  distance  from  the  point  left  to  the  point 
reached  ? 

Solve  the  same  problem  for  latitude  45°,  60°,  etc. 

Traverse  Sailing. 

135.  Traverse  Sailing  is  the  application  of  the  principles 
of  Plane  and  Middle  Latitude  Sailing  to  cases  when  the  ship 
sails  from  one  point  to  another  on  two  or  more  different 
courses.  Each  course  is  worked  by  itself,  and  these  inde- 
pendent results  are  combined,  as  may  be  seen  in  the  solution 
of  the  following  example  : 

136.  Leaving  latitude  37°  16'  S.,  longitude  18°  42'  W.,  a 
ship  sails  N.E.  104  miles,  then  KN.W.  60  miles,  then  W.  by 
S.  216  miles.  Find  the  position  reached,  and  its  bearing  and 
distance  from  the  point  left. 

We  have,  for  the  first  course,  difference  of  latitude  73.5  N., 
departure  73.5  E. 

We  have,  for  the  second  course,  difference  of  latitude, 
55.4  K,  departure  23  W. 

We  have,  for  the  third  course,  difference  of  latitude  42.1  S., 
departure  211.8  W. 

On  the  whole,  then,  the  ship  has  made  128.9  miles  of  north 
latitude,  and  42.1  miles  of  south  latitude.  The  place  reached 
is  therefore  on  a  parallel  of  latitude  86.8  miles  to  the  north  of 
the  parallel  left ;  that  is,  in  latitude  35°  49.2'  S. 

The  departure  is,  in  the  same  way,  found  to  be  161.3  miles 
W. ;  and  the  middle  latitude  is  36°  32.6'.     With  these  data. 


MISCELLANEOUS    EXAMPLES.  99 

and  the  formula  of  Example  123,  we  find  the  difference  of 
longitude  to  be  201',  or  3°  21'  W.  Hence  the  longitude 
reached  is  22°  3'  W. 

With  the  difference  of  latitude  86.8  miles,  and  the  departure 
161.3  miles,  we  find  the  course  to  be  N.  61°  43'  W.,  and  the 
distance  183.2  miles.  The  ship  has  reached  the  same  point 
that  it  would  have  reached,  if  it  had  sailed  directly  on  a 
course  N.  61°  43'  W.,  for  a  distance  of  183.2  miles. 

137.  A  ship  leaves  Cape  Cod  (Ex.  125),  and  sails  S.E.  by 
S.  114  miles,  N.  by  E.  94  miles,  W.KW.  42  miles.  Solve  as 
in  Ex.  136. 

138.  A  ship  leaves  Cape  of  Good  Hope  (latitude  34°  22'  S., 
longitude  18°  30'  E.),  and  sails  N.W.  126  miles,  N.  by  E.  84 
miles,  W.S.W.  217  miles.     Solve  as  in  Ex.  136. 

Problems  in  Goniometry. 
Prove  that 

1.  sin  X -|- cos  £c  =  V2  cos  (x  —  Jtt). 

2.  sinic  — cos  £c  =  —V2  cos  (cc  +  iTr). 

3.  sina;+ V3cosic  =  2sin(a?  +  -J-7r). 

4.  sin  (x  + -J-tt)  +  sin  (cc  —  •J7r)  =  sinx. 

5.  cos  (^  +  ^  tt)  +  cos  (x  —  ^  7r)  =  V3  cos  X, 

6.  tancc  +  seca:  =  tan(icc4-:|^7r). 
1 


7.   tan  X  -\-  sec  x 
8. 


sec  X  —  tan  x 
1  —  tan  X      cot  x—1 


1  +  tan  x      cot  x-\-l       ■ 

sin  X       ,  1  +  cos  X 

9.   zr-x -• =  2csca;. 

1  +  cos  X  sm  X 

10.  tan£c  +  cotcc  =  2csc2ic.     12.  1 +tan£ctan2a;  =  sec2x. 

sec  cc 

11.  cotic  — tana3  =  2cot2ic.     13.  sec2ic  =  ^r 5— 

2  —  sec^ic 


100  TRIGONOMETRY. 

Prove  that 

14.    2sec2a^  =  sec(cc  +  45°)sec(a;  — 45°). 
cos  X  -\-  sin  X 


15.    tan  2  a;  +  sec  2  a:;  = 


cos  X  —  sm  X 


•    ^  2  tan  a?  ^rr     o    •         .     •    o  2sin^a7 

16.    sm 2a;— ■    ,   . — n—         17.    2sina;4-sin2a; 


18.    sin  3  a;  = 


1  +  tan^a;  1  —  cos  a; 

sin^  2  a;  —  sin^a; 


sma; 

3  tan  a;  —  tan^x  tan  2a;  +  tan  a; sin  3a; 

1  —  Stanza;  *  tan  2a;  — tana;        sin  a; 

21.  sin  (x-\-y)-\-  cos  (x  —  y)  =  '^  sin  (a;  +  :J-7r)  sin  {y  -\-  Jtt). 

22.  sin  (x-{-y)  —  cos  (a;  —  ?/)  =  —  2  sin  (x  —  ^ir)  sin  {y  —  ^  tf). 

23.  tanx  +  tany^^'"(^+y>- 

COS  a;  cos  3/ 

^.     ^      /     .     N      sin  2a;  +  sin  2y 

24.  tan  (a; +  2/)= ^^ — ^• 

^        "^^      cos  2  a;  4- cos  2?/ 

2^     sin  a;  +  cosy  _  tan  j  j-(a;+?/)4-45°^ 
sin  a;  —  cos?/      tanJ-J(a;  —  y)  —  45°  J 

26.  sin  2a;  +  sin  4a;  =  2  sin  3a;  cos  X. 

27.  sin  4  a;  =  4  sin  a;  cos  a;  —  8  sin^a;  cos  x 

=  8  cos^a;  sin  x  —  4  cos  x  sin  x. 

28.  cos  4  a;  =  1  —  8  cos^a;  +  8  cos^a;  =  1  —  8  sin^ic  +  8  sin*a;. 

29.  cos  2  a;  +  cos  4  a;  =  2  cos  3  a;  cos  a;. 

30.  sin  3a;  —  sin  a;  =  2  cos  2a;  sin  a;. 

31.  sin^a;  sin  3  a;  +  cos^a;  cos  3  a;  =  cos'  2  x. 

32.  cos^a;  —  sin*a;  =  cos  2  x. 

33.  cos^a;  -f-  sin*a;  =  1  —  ^  sin^  2  x. 

34.  cos^a;  —  sin^a;  =  cos  2  a;  (1  —  sin^a;  cos^a;). 

35.  cos^a;  -j-  sin^a;  =  1  —  3  sin^a;  cos^x. 

„„     sin  3  a;  4"  sin  5  a; 

06. — -  =  cot  x. 

cos3x  —  cos  5a; 


MISCELLANEOUS    EXAMPLES.  101 


Prove  that 


^„    Sin3a7  +  sm5x 

37.  —. r— 7— 7--  =  2cos2ic. 

smx  +  sm3a; 

38.  csccc  —  2 cot 2 cc cos ic^ 2 since. 

39.  (sin  2  cc  —  sin  2  ?/)  tan  (cc  -j-  ?/)  =  2  (sin^aj  —  sin^i/). 

sec  07       csccc 


csc^a?       sec^ic 
sin2  3£c 


40.  (1  +  cot  £c  4"  tan  x)  (sin  x  —  cos  x) 

41.  sina;  +  sin3cc  +  sin5ic  = 

smic 

,^    3cosic  +  cos3ic         ,, 

42.  77-^ ■ — r— :— =  cot^aj. 

3sinic  —  smSic 

43.  sin  3  ic  =  4  sin  x  sin  (60°  +  x)  sin  (60°  —  x). 

44.  sin4a;  =  2siniccos3ic  +  sin2ic. 

45.  sinic  +  sin(£c  — f  7r)4-sin(^7r  — a7)  =  0. 

46.  cos X sin (?/  —  ^)  +  cos y sin (z  —  x)-\- cos z sin (x  —  y)=^ 0. 

47.  cos  (x  +  2/)  sin  y  —  cos  (x  +  z)  sin  « 

=  sin  (x  -\-  y)  cos  ?/  —  sin  (x  +  ^)  cos  z. 

48.  cos  (cc  +  ?/  +  ^)  -f-  cos  (cc  +  2/  —  «)  +  cos  (x  —  y-}-z) 

+  cos  (2/  +  «  —  cc)  ^  4  cos  X  cos  y  cos  «. 

49.  sin  (x  +  ?/)  cos  (ic  —  ?/)  +  sin  (y  -\-  z)  cos  {y  —  z) 

+  sin  {z  +  a?)  cos  (^  —  a;)  =  sin  2 x  -|-  sin  2y-\-  sin  2^. 
60.   si°7y  +  siniy      ^^^ 
sin75°  — sinl5° 

51.  cos  20°  + cos  100°  + cos  140°  =  0. 

52.  cos  36°  +  sin  36°  =V2  cos  9°. 

53.  tan  11°  15'  +  2  tan  22°  30'  +  4  tan  45°  =  cot  11°  15'. 

If  A,  B,  C  are  the  angles  of  a  plane  triangle,  prove  that 

54.  sin  2^  +  sin  2^  +  sin  2  C  =  4  sin^  sin^ sin  C. 

55.  cos  2  ^  +  cos  2  .B  +  cos  2  C  =  —  1  —  4  cos  A  cos  B  cos  C. 


102 


TRIGONOMETRY. 


If  A,  B,  C  are  the  angles  of  a  plane  triangle,  prove  that 

56.  sin 3^  + sin 3^+ sin 3  C  =  — 4cos  -;^cos  -^  cos  -^' 

Z  Z  2i 

57.  Gos^A  +  cos^-B  +  cos^  C  =  1  —  2  cos  A  cos  B  cos  C. 

If  A^B+C  =  90°,  prove  that 

58.  tan^  tan J5  +  tan^  tan  C  +  tan  C  tsmA  =  1. 

59.  sin^^  +  sin2^  +  sin^  (7  =  1  —  2  sin^  sin  jB  sin  C. 

60.  sin  2^  +  sin  2^  +  sin  2  C  =  4  cos^  cos  J5  cos  a 
Prove  that 


61.  sin  (sin-^  x  +  sin~^  y)  z=x  \ll  —  if  -\-  ysl  1  —  x\ 

62.  tan  (tan-^ x  +  tan-^^/)  =  ^^^- 

63.  2tan-ix  =  tan-i:r^^- 

1—x^ 


64.  2  sin-^a;  =  sin-i  (2a;  Vl  —  x^, 

65.  2cos-^x  =  cos-i(2ic2  — 1). 
3£c  — a;^ 


QQ.    3  tan~^  x  =  tan~^ 


1-3x2 


67.  sin-iV-  =  tan-i\/ 

68.  sin-i\/^^^  =  tan-i-\/^::^. 


y  —  x 


l-2x  +  4x2  '  —    l  +  2aj+4x2-^'^ii    2a;'' 
1 


69.  tan-i -— — — — -^  +  tan-i  :7-n-7^ — r-r-5=  tan 

70.  sin-^a;  =  sec~^ 


Vl  — a;2 


71.  2sec-^a;  =  tan-^^y^'~^. 

2— or 

72.  tan-ii  +  tan-ii  =  45°. 


MISCELLANEOUS    EXAMPLES.  103 

73.  tan-i  ^  _|_  ^^^j^-i  ^  _  ^^j^-i  ^^ 

74.  sin-^  I H-  sin-^  ||  =  sin-^  f  |. 

75.  sin-i-^+siii-i-4^=45°. 

V82  V41 

76.  sec-i  I  +  sec-i  ||  ^  75°  45'. 

77.  tan-i  (2  +  V3)  —  tan-^  (2  —  V3)  =  sec-^  2. 

78.  tan-i  ^  _^  ^^j^-i  ^  _^  ^an-i  i  +  tan-^  ^  =  45°. 

79.  Given  cos  ic  =  f ,  find  sin  ^cc  and  cos  ^x, 

80.  Given  tan  x  =  ^,  find  tan  ^x. 

81.  Given  sin  x  -\-  cos  a:  =  V^,  find  cos  2x. 

82.  Given  tan  2x  =  \^- ,  find  sin  x. 

83.  Given  cos  3  x  =  |f ,  find  tan  2  a;. 

84.  Given  2  cscx  — cotcc=  V3,  find  sin -J a;. 

85.  Find  sin  18°,  cos  36°. 
Solve  the  following  equations  : 

86.  sin£c  =  2sin  (-J-TT  +  a;).         90.    sinic  +  cos2ic  =  4sin2a;. 

87.  sin  2ic  =  2cosa7.  91.   4  cos  2a; +  3  cos  a?  =  1. 

88.  cos  2  £c  =  2  sin  a;.  92.    sin  a;  +  sin  2  a;  =  sin  3  a;. 

89.  sin  a;  +  cos  a;  =  1.  93.    sin  2a;:=3sin^a;  —  cos'^a;. 

94.  tan  a;  +  tan  2  a;  =  tan  3  a;. 

95.  cot  a;  —  tanx=:sina;  +  cosa;. 

96.  tan^a;  =  sin2a;.  99.   sina;+sin2a;=l — cos2a:. 

97.  tana;  +  cota;:=tan2a;.       100.    sec2a^  +  l  =  2cosa;. 

98.  -^~l^'^^  =  cos2x.  101.   tan 2a;  +  tan  3a;  =  0. 
1  +  tan  X 


102.  tan(i7r  +  a;)+tan(i7r  — a;)  =  4. 

103.  Vl  +  sina;  —  Vl  —  sina;  =  2cosx. 


104  TRIGONOMETRT. 

Solve  the  following  equations  : 

104.  tanxtan3ic  =  — f. 

105.  sin(45°+ic)  +  cos(45°  — ic)  =  l. 

106.  tSinx-{-seGx  =  a.  107.   cos2a;  =  a(l  — coscc). 

108.  cos2£c  (1  — tancc)  =  a(l-[-tanic). 

109.  sin^  X  +  cos^x  =  j\  siu^  2  x. 

110.  cosScc  +  Scos^x^O. 

111.  sec  (x  + 120°)  +  sec  (x  — 120°)  =  2  cos  x. 

112.  cscic  =  cota:4- V3.  114.    coscc  — cos  2a;  =  l. 

113.  4cos2ic  +  6sinic  =  5.      115.    sin  4  a?  —  sin  2  a:  =  sin  ic. 

116.  2sin2x  +  sin2  2x  =  2. 

117.  cos5x  +  cos3a3  +  cosa:  =  0. 

118.  secic  —  cot  x  =  CSC  a:  —  tana;. 

119.  t&-n.^x-i-cot^x  =  i^^-. 

120.  sin4a;  — cos3a;  =  sin2a;. 

121.  sina;  +  cosa;  =  secx.  122.   2cosa;cos3a^  +  l  =  0. 

123.  cos3a;  — 2cos2a;  +  cosx  =  0. 

124.  tan  2  a;  tan  a:  :=1. 

125.  sin  (x  + 12°)  +  sin  (x  —  8°)  =  sin  20°. 

126.  tan(60°  +  a;)tan(60°  — a;)  =  — 2. 

127.  sin  (a; +  120°)  + sin  (a;  +  60°)  =  1. 

128.  sin(a;  +  30°)sin(a;  — 30°)==i. 

129.  sin*a;  +  cos*a;  =  |.  131.    tan  (a;  +  30°)  =  2  cos  cc. 

130.  sin'^a;  —  cos^x  =  ^5^.  132.    sec  a;  =  2  tana;  +  J. 

133.  sin  (x  —  y)=  cos  x,  cos  (x-{-y)  =  sin  x. 

134.  tan  x  +  tan  y  =  a,    cot  x  +  cot  y  =  b. 

135.  sin  (a; +  12°)  cos  (a;  — 12°)=  cos  33°  sin  57°. 

136.  sin-^ X  +  sin-i  ix  =  120°. 

137.  tan-ia;  +  tan-i2a;  =  tan-i3V3. 

138.  sin-ix  +  2cos-ia;  =  |7r. 


MISCELLANEOUS    EXAMPLES.  105 

Solve  the  following  equations  : 

139.  sin-ia;  +  3cos-i£c  =  210°. 

140.  tan-ia;  +  2cot-iaj  =  135°. 

141.  tan-i  (^  _j_  1)  _|.  tan-i  (^^_i^^  tan-^  2  x. 

142.  tan-i^  +  tan-i^^^fTT. 

x-\-l  X  —  1 

143.  tan-i3^^„-::60°. 

1  —  x^ 

Find  the  value  of  : 

144.  asecic  +  ^cscic,  when  tancc^AZ- 


145.  sin  3  x,  when  sin  2  cc  =  Vl  —  ml 

pso  iT*  ^"^  sec  cc  / — 

146.  — 5 — i r- ,  when  tan  x  =  \l\. 

147.  sin  X,  when  tan^cc  +  3  cot^cc  =  4. 

148.  cos  X,  when  5  tan  x  +  sec  a?  =  5. 

a 

149.  sec  07,  when  tan  x  = 


V2a+1 

Simplify  the  following  expressions  : 
(cos  X  +  cos  ijY  +  (sin  x  +  sin  y)' 

cos'^  i  (»^  —  2/) 
sin  (a; H-  2y)  —  2  sin  (x-^y)-\-  sin .t 
cos  (.T  4- 2?/)— 2  cos  (£c  +  2/)  +  cosa; 
sin  (g;  —  ^)  +  2  sina;  4-  sin  (x  +  g) 
*    sin(?/  — ;^)  +  2sin2/+sin(2/+2;) 

cosGa?  —  cos  4a; 

153.  -: — -^ — j — : — -. — 

sm  6a;  +  sin4a7 

154.  tan-i  (2a;  + 1)  +  tan-i  ^2x- 1). 


1 


1^^-    1  4_  sin2  ^  +  1  -I-  cos^a;  "^  1  +  sec^x  "^  1  +  csc^x 
156.    2sec2a;  — sec*a3  — 2csc2a:;  +  csc'*a;. 


EE"TEA]^OE  EXAMII^ATIOK"  PAPEES.* 


PLANE   TRIGONOMETRY  AND  LOGARITHMS. 


{Cornell,  June,  1889.) 
(One  question  may  be  omitted.) 

1.  Prove  that 

cos  co-^  =  sin  ^  ; 

sec  (^V  +  ^)  =  —  CSC  ^ ; 
tan  (—0)=  —  tan  6  j 

CSC  (7r  —  0)=  CSC  $. 

2.  Draw  the  curve  of  tangents,  and  show  the  changes  in 
the  value  of  this  function  as  the  arc  increases  from  0°  to  360°. 

3.  In  terms  of  functions  of  positive  angles  less  than  45°, 
express  the  values  of  sin  — 250°,  esc  j|7r,  tan  —  J/  tt.  Also 
find  all  the  values  of  6  in  terms  of  a  when  cos  ^  =  Vsin^a. 

4.  (a)  Given  cos  x  =  0.5,  find  cos  2x  and  tan  2ic. 

(b)  Prove  that  vers  (180°  —  A)-\-  vers  (360°  —  A)  =  2. 

5.  Prove  the  check  formulae  : 

a-]rb  :  c  =  cos^  (A  —  B)  :  sin^  C; 
a  —  b:c  =  sini(A  —  B):cosiC. 

*  Note.  In  these  papers,  as  in  many  text-books,  the  Greek  letters 
a  (alpha),  ^  (bayta),  y  (gamma),  5  (delta),  6  (thayta),  and  0  (phee),  are 
occasionally  used  to  denote  angles. 


ENTRANCE  EXAMINATION  PAPERS.  107 

6.  In  a  right  triangle,  r  (the  hypotenuse)  is  given,  and  one 
acute  angle  is  n  times  the  other ;  find  the  sides  about  the  right 
angle  in  terms  of  r  and  n. 

7.  The  tower  of  McGraw  Hall  is  125  ft.  high,  and  from  its 
summit  the  angles  of  depression  of  the  bases  of  two  trees  on 
the  campus,  which  stand  on  the  same  level  as  the  Hall,  are 
respectively  57°  44'  and  16°  59',  and  the  angle  subtended  by 
the  line  joining  the  trees  is  99°  30'.  Find  the  distance  between 
the  trees. 


11. 

{Cornell^  June,  1890.) 
(Omit  one  question.) 


1.  Prove  that  cot  ( —  0)  =  —  cot  6  ;  esc  tt  —  0  =  esc  6  ; 
sin(7r4-^)=  — sin^;  sec  co-^  =  esc  ^  ;  cos  (^7r-\-6)=^  —  sm6. 

2.  Show  that  in  any  plane  triangle  sin  iA  =  ^ ^ ^• 

3.  Find  the  value  of  sin  (6  ±  0')  in  terms  of  sin  6,  cos  6, 
sin  O'j  and  cos  0'. 

4.  Given  tan45°  =  l ;  find  all  the  functions  of  22°  30'. 

5.  Determine  the  number  of  solutions  of  each  of  the  tri- 
angles: a  =  13.4,  ^»  =  11.46,  A  =  77°20'',  c  =  58,  a=75,  0=60°^ 
b  =  109,  a  =  9A,  A  =  92°10';  c  =  S09,  b  =  360,  (7=21°  14'25". 

6.  In  a  parallelogram,  given  side  a,  diagonal  d,  and  the 
angle  A  formed  by  the  diagonals  ;  find  the  other  diagonal  and 
the  other  side. 

7.  A  and  B  are  two  objects  whose  distance,  on  account  of 
intervening  obstacles,  cannot  be  directly  measured.  At  the 
summit  of  a  hill,  whose  height  above  the  common  horizontal 


*-^-.^ 


108  TKIGONOMETRY. 

plane  of  the  objects  is  known  to  be  517.3  yds.,  angle  ACB  is 
found  to  be  15°  13'  15".  The  angles  of  elevation  of  C  viewed 
from  A  and  B  are  21°  9'  18"  and  23°  15'  34"  respectively.  Pind 
the  distance  from  A  to  B. 


III. 

{Cornell,  September,  1891.) 

1.  Trace  the  value  of  tan  0  and  that  of  esc  6,  as  0  increases 
from  0°  to  360°. 

2.  (a)  Find  the  remaining  functions  of  ^  when  cos  6  = — |-  V3. 
(b)  Determine  all  the  values  of  $  that  will  satisfy  the 

relation  cot  ^  —  2  cos  0. 

3.  Prove  the  identity 

tan  ^  —  cot  ^  =  — : — y~  =  —  2  cot  2  A. 

sm  ^  cos  ^ 

4.  Derive  an  expression  for  the  sine  of  half  an  angle  in  a 
triangle  in  terms  of  the  sides  of  the  triangle. 

5.  Construct  a  figure  and  explain  fully  (giving  formulae) 
how  you  would  find  the  height  above  its  base,  and  the  distance 
from  the  observer,  of  an  inaccessible  vertical  object  that  is 
visible  from  two  points  whose  distance  apart  is  known,  and 
which  can  be  seen  from  one  another. 

6.  Given  two  sides  of  a  plane  triangle  equal  respectively  to 
121.34  and  216.7,  and  the  included  angle  47°  21' 11",  to  find 
the  remaining  parts  of  the  triangle. 

7.  In  a  right  triangle,  if  the  difference  of  the  base  and  the 
perpendicular  is  12  yds.,  and  the  angle  at  the  base  is  38°  1'  8", 
what  is  the  length  of  the  hypotenuse  ? 


ENTRANCE  EXAMINATION  PAPERS.  109 

IV. 

{Cornell,  June,  1892.) 

1.  By  means  of  an  equilateral  triangle,  one  of  whose  angles 
is  bisected,  find  the  numerical  values  of  the  functions  of  30° 
and  60°. 

2.  If  6  be  any  angle,  prove  that 

sin  6  =  tan  ^  :  Vl  +  tan^  0,     cos  6  =  Vcsc^  9  —  1:  esc  0. 

3.  Prove  that  ^^^^+^^^^  =  -  cot  ^  (6-0'),  where  6  and  0' 

cos  ^  —  cos  ^'  ^  ^         ^' 

are  any  angles. 

4.  Find  sin  2  ^,  cos  2  0,  and  tan  2  ^,  in  terms  of  functions  of  0. 

5.  Assuming  the  law  of  sines  for  a  plane  triangle,  prove  that 

(a-\-b)  :  c  =  cosi(A  —  B)  :  sin ^  C, 
(a  —  b)  :  c  =  sin  ^(A  —  B)  :  cos  -J-  C. 

6.  At  120  feet  distance,  and  on  a  level  with  the  foot  of  a 
steeple,  the  angle  of  elevation  of  the  top  is  62°  27';  find  the 
height. 

7.  Solve  the  plane  triangle  given  the  three  sides, 

a  =  48.76,  ^'^  62.92,  c  =  80.24. 


V. 

{Harvard,  June,  1889.) 

1.  In  how  many  years  will  a  sum  of  money  double  itself  at 
4  per  cent.,  interest  being  compounded  semi-annually  ? 

2.  Given  sin^a?  = ?  find  sin  2x  and  tan  2  x. 

z 

3.  Find  all   values   of  x,  under  360°,   which  satisfy  the 
equation  V8  cos  2  ic  =  1  —  2  sin  ic. 


110  TRIGONOMETRY. 

4.  What  is  always  the  value  of 

2  sin^ic  sin^y  +  2  cos^a;  cos^y  —  cos  2x  cos  2y? 

5.  Find  the  area  of  a  parallelogram,  if  its  diagonals  are 
2  and  3,  and  intersect  each  other  at  an  angle  of  35°. 

6o   Find  the  bearing  and  distance  from  Cape  Horn  (55°  55'  S., 
67°  40'  W.)  to  Falkland  Island  (51°  40'  S.,  59°  W.). 


VI. 


{Earmrd^  June,  1890.) 

1.  In  a  certain  system  of  logarithms  1.25  is  the  logarithm 
of  -J.     What  is  the  base  ? 

Be  careful  to  remember  what  1.25  means. 

2.  Find  the  tangent  of  3  a;  in  terms  of  the  tangent  of  x. 

3.  One  angle  of  a  triangle  is  35°,  and  one  of  the  sides 
including  this  angle  is  24.  What  are  the  smallest  values  the 
other  sides  can  have  ? 

4.  Find  all  values  of  ic,  under  360°,  which  satisfy  the 
equation 

tan  2x  (tSLTi^x  —  1)  =  2  sec^x  —  6. 

5.  Two  ships  leave  Cape  Cod  (42°  N.,  70°  W.),  one  sailing 
E.,  the  other  sailing  N.E.  How  many  miles  must  each  sail  to 
reach  longitude  65°  W.  ? 

6.  UA-{-B+C  =  180°,  find  the  value  of 

« 
tan  A  -\-  tan  B  +  tan  C  —  tan  A  tan  B  tan  C. 


ENTRANCE    EXAMINATION    PAPERS.  Ill 

VII. 

{Harvard,  September,  1891.) 

1.  What  is  the  base,  when  log 0.008  =  —  1.5? 

2.  If  cos  (a—b)  =  3  cos  (a-\-b),  find  the  value  of I  • 

^        ^  ^         ^  sec  a  sec  b 

3.  The  area  of  an  oblique-angled  triangle  is  50.  One  angle 
is  30°,  and  a  side  adjacent  to  that  angle  is  12.  Solve  the 
triangle. 

4.  Find  all  values  of  x,  less  than  360°,  which  satisfy  the 
equation 

sin  2x  —  cos  x  =  cos^x. 

5.  Find,  by  Middle  Latitude  Sailing,  the  course  and  the 
distance  from  Cape  Cod  (Lat.  42°  2'  N.,  Long.  70°  4'  W.)  to 
Fayal  (Lat.  38°  32'  N.,  Long.  28°  39'  W.). 

6.  In  any  triangle  ABC,  prove 

tan  ^  A  tan  ^B  -\-  tan  i  A  tan  ^  C  +  tan  -J  B  tan  -J-  C  =  1. 


VIIL 

{Harvard,  September,  1892.) 

(Take  the  questions  in  any  order.     One  of  the  starred  questions  may  he 

omitted.) 

1.   What  is  the  base  of  a  system  of  logarithms  in  which 
logGi^)  =  2.33i? 

*2.    Given  the  area  of  a  right  triangle,  and  the  smallest 
angle,  find  the  legs  of  the  triangle  in  terms  of  the  data. 

^,^     -r^.    -,  -,  ^     .         sina         /-        ,  tana         ,- 

=^3.    Find  a  and  B,  given  -7—7:  =  V2,  and ^  =  V3. 

^'  *  sm/3  tan/? 


9 

i 


112  TRIGONOMETRY. 

4.  One  angle  of  an  oblique-angled  triangle  is  45°,  and  an 
adjacent  side  is  V2.  What  is  the  smallest  value  which  the 
opposite  side  can  have  ?  Solve  the  triangle  when  the  opposite 
side  is  |. 

5.  A  ship  leaves  Cape  Cod  (42°  2'  N.,  70°  4'  W.)  and  sails 
200  knots  on  a  course  S.  40°  E.  Find  the  latitude  and  longi- 
tude reached. 

*6.    If  2  tan  2a  =  tan  25  sin  2  J,  find  the  relation  between 
the  tangents  of  a  and  k 


IX. 


{Harvard,  June,  1893.) 

(Take  the  problems  in  any  order.     One  of  the  starred  problems  may  be 

omitted.) 

1.   What  is  the  base  of  the  system  of  logarithms,  when 
log  3  =  0.3976? 

*2.  Solve  the  right-angled  triangle  in  which  one  angle  is 
30°,  and  the  difference  of  the  legs  is  4. 

=^3.    Find  x,  given  sec  a;  =  2  tan  x-\-2. 

*4.  One  angle  of  a  triangle  is  double  another  angle.  The 
side  opposite  the  first  angle  is  three-halves  of  the  side  opposite 
the  second  angle.     Find  the  angles. 

tV"'^  6.  Find,  by  Middle  Latitude  sailing,  the  course  and 
distance  from  Funchal  (32°  38'  N.,  16°  54'  W.)  to  Gibraltar 
(36°  7' K,  5°  21' W.). 

*6.   Eeduce  to  its  simplest  form  cos  2  x  tan  (45°  +  ic)  —  sin  2  x. 


ENTRANCE  EXAMINATION  PAPERS.  113 

{Harvard,  September,  1893.)  * 

(One  of  the  starred  problems  may  be  omitted.) 

1.  If  the  base  of  our  system  of  logarithms  were  20  instead 
of  10,  what  would  be  the  logarithm  of  one  tenth? 

*2.   The  area  of  a  right  triangle  is  6,  and  the  sum  of  the 
three  sides  is  12.     Solve  the  triangle, 

*3.   Eeduce  to  its  simplest  form 

cos2J5  +  sin2^cos2^  — sin2^cos2^. 

*4.    Two  angles  of  a  triangle  are  40°  14'  and  60°  37'.     The 
sum  of  the  two  opposite  sides  is  10.     Find  these  sides. 

5.  A  ship  leaves  Cape  of  Good  Hope  (34°  22'  S.,  18°  30'  E.), 
and  sails  K  40°  W.  to  Latitude  30°  S.  Find,  by  Middle  Lati- 
tude  Sailing,  the  Longitude  reached  and  the  distance  sailed. 

*6.    The  base  angles  of  a  triangle  are  22°  30'  and  112°  30'. 
Find  the  ratio  between  the  base  and  the  height  of  the  triangle. 


{Harvard,  June,  1894.) 
(Arrange  your  work  neatly.) 

1.  What  is  meant  by  the  logarithm  of  a  number  n  in  the 
system  whose  base  is  8  ?    What  will  be  the  logarithm  of  4  in 

this  system? 

2.  Establish  the  formula  :  

,^    ,  ^          N     /l  —  cos  a; 
sin|x  =  ±(l+2cosa;)-Y 

Which  sign  should  be  used  when  x  lies  in  the  first  quadrant  ? 
When  X  lies  in  the  second  quadrant? 


114 


TRIGONOMETRY. 


3.  In  a  triangle  two  angles  are  equal  to  32°  47'  and  49°  28' 
respectively  and  the  length  of  the  included  side  is  0.072. 
Solve  the  tftangle. 

4.  A  circular  tent  30  feet  in  diameter  subtends  at  a  certain 
point  an  angle  of  15°.  Find  the  distance  of  this  point  from 
the  centre  of  the  tent. 

5.  A  ship  leaves  Latitude  42°  2'  N.,  Longitude  70°  3'  W., 
and  sails  K  40°  E.  a  distance  of  420  miles.  Find  by  Middle 
Latitude  Sailing  the  position  reached. 


XII. 

{Sheffield  Scientific  School,  September,  1892.) 

1.  Express  an  angle  of  60°  in  radians. 

2.  Kepresent  geometrically  the  different  trigonometric 
functions  of  an  angle.  State  the  signs  of  each  function  for 
each  quadrant. 

3.  Express  tan  <^  and  sec  <^  in  terms  of  sin  <^. 

4.  Derive  the  formula 

sin  a  +  sin  /3  =  2  sin  ^  (a-f-  ^)  cos  i  (a  —  (3). 

5.  Show  that,  if  a,  h  and  c  are  the  sides  of  a  triangle  and 
A  is  the  angle  opposite  the  side  a,  then  a^=h'^-[- 0^—21)6  cos  A. 

6.  Given  cos  2  a?  =  2  sin  x,  to  find  the  value  of  sin  x. 

7.  Given  two  sides  of  a  triangle  a  =  450.2,  ^  =  425.4,  and 
the  included  angle  C  =  62°  8'j  find  the  remaining  parts. 


xin. 

(Sheffield  Scientific  School,  June,  1893.) 

1.  Express  an  angle  of  15°  in  radians. 

2.  Write  the  simplest  equivalents  for  sin  (7r+<^),  tan  (27r — cf>). 

C0S(|7r— «^),  Sec(7r+<^). 


ENTRANCE  EXAMINATION  PAPERS.  115 

3.  Express  tan  (f>  in  terms  of  sin  (f>,  cos  <^  and  cot  <f>,  respect- 
ively; and  cos  <f>  in  terms  of  tan  <^,  sec  </>  and  cosec  <f>,  respectively. 

4.  Show  (a)  that  sin  (a  +  y8)  +  sin  (a  —  y8)  =  2  sin  a  cos  y8 ; 

(b)  that  cos  (a + /?)  +  cos  (a  —  )8)  =  2  cos  a  cos  13. 

5.  Assume  the  formula  cosa= tt-, and  show  that 

2  00 

sin^-J-a=  ^^ y^^ -,  when  s^i(a-\-b-\-c). 

oc 

6.  Obtain  a  formula  for  tan  -J  a  in  terms  of  cos  a. 

7.  The  base  of  a  triangle  c  =  556.7,  and  the  two  adjacent 
angles  a  =  65°  20.2',  /8  =  70°  00.5';  calculate  the  area  of  the 
triangle. 

8.  Given  0  <  a  <  90°,  and  log  cos  a  =  1.85254,  to  determine  a. 


XIV. 

(Sheffield  Scientific  School,  September,  1893.) 

1.  Eeduce  an  angle  of  3.5  radians  to  degrees. 

2.  Define  the  different  trigonometrical  functions  of  an 
angle  and  give  their  algebraic  signs  for  an  angle  in  each 
quadrant. 

3.  Write  simple  equivalents  for  the  following  functions  : 
sin  (—a);     cos  (—a);     tan(|7r  +  a);     sec(|7r  — a). 

4.  Express  cosec  a  in  terms,  respectively,  of  sin  a,  cos  a, 
tan  a,  cot  a,  sec  a. 

5.  Reduce 

(cos  a  cos  /?  —  sin  a  sin  ^8)^+  (sin  a  cos  /?  +  cos  a  sin  py 
to  its  simplest  equivalent. 


»     r^-,        .^    .  .      ( "^        \      1— tan  a 

6.    Show  that  tan    y  —  a    =  .    ,  ^ 

V4        y      1  +  tan  a 


116  TRIGONOMETRY. 

7.  The  sum  of  two  sides,  a  and  h,  of  a  triangle  is  546.7  ft., 
tlie  sum  of  the  opposite  angles,  a  and  p,  is  124°,  and  sin  a:  sin  j8 
=  1.003 ;  find  the  angles  and  sides  of  the  triangle. 

8.  Given  0  <  a<  90°,  and  log  cot  a  =  0.03293,  to  determine  a. 


XV. 

{Sheffield  Scientific  School,  June,  1894.) 

1.  Express  (a)  an  angle  of  2  radians  in  degrees ; 

(h)  an  angle  of  30°  in  radians. 

2.  Give  simple  equivalents  for  the  following  functions : 
tan  (—  x),  cosec  (—  x),  sin  {x-\-\ tt),  sin  («  —  ^  tt),  tan  (| tt  —  ic), 
sin(27r  —  x). 

3.  Given  tan  a;  =  -?  to  express  sin  x,  cos  x,  cot  ic,  sec  x,  and 
cosec  X  in  terms  of  a  and  b, 

.     ct,        ,,    ,  ,  ,       ,       sin  (a  dtzb) 

4.  (Show  that  tan  a  ±  tan  b  = ^ 7* 

cos  a  cos  0 

6.   Derive  the  formulae 


,                /l  +  cosa      .     ,  / 

cosia  =  ±yl ,  sin|-a  =  ±-y 


1  •—  cos  a 


2        '         ^  -      -  \        2 

6.  Given  180°  <  </>  <  270°,  and  log  cot  <^  =  0.03232,  find  4,. 

7.  The  sides  of  a  triangle  are  a  =  32.5  ft.,  J  =  33.1  ft., 
c  =  32.4  ft. :  calculate  the  area  of  the  triangle  and  the  angle  C 
opposite  the  side  c,  using  the  following  formulae : 


S='^p(p  —  a)(p  —  b)  (p  —  c)  =  iab  sin  C, 
in  which  S  denotes  the  area  of  the  triangle,  smd  p=i(a-\-b-\-c). 


CHAPTER   VI. 

CONSTRUCTION   OF   TABLES. 

§  42.    Logarithms. 

Properties  of  Logarithms.  Any  positive  Dumber  being 
selected  as  a  base,  the  logarithm  of  any  other  positive  number 
is  the  exponent  of  the  power  to  which  the  base  must  be  raised 
to  produce  the  given  number. 

Thus,  if  a"  =  JV,  then  n  =  logaJV. 

This  is  read,  n  is  equal  to  log  N  to  the  base  a. 

Let  a  be  the  base,  M  and  N  any  positive  numbers,  m  and  n 
their  logarithms  to  the  base  a ;  so  that 

a'^  =  M,  a''  =  N, 

Then,  in  any  system  of  logarithms : 

1.  The  logarithm  of  1  is  0. 

For,  a«  =  l.  .•.0  =  log„L 

2.  The  logarithm  of  the  base  itself  is  1. 
For,  a^  =  a.  .•.l=log„a. 

3.  The  logarithm  of  the  reciprocal  of  a  positive  number  is  the 
negative  of  the  logarithm  of  the  number. 

For,  if  a"  =  iV,  then  —=  —  =  «-». 

'  '  N      a"" 


(^)— 


logahv?      =-^  =  -10g„iV. 


H^  TRIGONOMETRY. 

4.  The  logarithm  of  the  product  of  two  or  more  positive 
numbers  is  found  by  adding  together  the  logarithms  of  the 
several  factors. 

For,  jf  xiV^==a"*Xa«  =  a'"  +  ^  ' 

Similarly  for  the  product  of  three  or  more  factors. 

5.  The  logarithm  of  the  quotient  of  two  positive  numbers 
is  found  by  subtracting  the  logarithm  of  the  divisor  from  the 
logarithm  of  the  dividend. 

J?Or,  zz= —  fjm  —  n 

jsr     a-      ""      ' 

6.  The  logarithm  of  a  power  of  a  positive  number  is  found  by 
multiplying  the  logarithm  of  the  number  by  the  exponent  of 
the  power. 

For,  ]srP  —  (^a''y  =  a''P. 

7.  The  logarithm  of  the  real  positive  value  of  a  root  of  a 
positive  number  is  found  by  dividing  the  logarithm  of  the 
number  by  the  index  of  the  root. 

For,  7]i^=7^^^7 

r  r 

Change  of  System.      Logarithms   to   any  base  a  may  be 
converted  into  logarithms  to  any  other  base  b  as  follows  : 
Let  iVbe  any  number,  and  let 

n  =  log„  iV^  and  m  =  logj,  iV. 
Then,  ]sr=  a"  and  ]V=  b"". 

.-.  a"  =  b"^. 


CONSTRUCTION    OF    TABLES.  119 

Taking  logarithms  to  any  base  whatever, 
n  log  a=^m  log  b, 
or,  log  a  X  log„  N^  log  b  X  log^  N, 

from  which  log^,  N  may  be  found  when  log  a,  log  b,  and  log^  N 
are  given ;  and  conversely,  log„  N  may  be  found  when  log  a, 
log  &,  and  logb  N  are  given. 

Two  Important  Systems.  Although  the  number  of  different 
systems  of  logarithms  is  unlimited,  there  are  but  two  systems 
which  are  in  common  use.     These  are : 

1.  The  common  system,  also  called  the  Briggs,  denary,  or 
decimal  system,  of  which  the  base  is  10. 

2.  The  natural  system  of  which  the  base  is  the  fixed  value 
which  the  sum  of  the  series 

1    1   1   1     1     I      1      I        1        .        ■ 
"^  1  "^  1.2  "^  1.2.3  "^  1.2.3.4  "^ 

approaches  as  the  number  of  terms  is  indefinitely  increased. 
This  fixed  value,  correct  to  seven  places  of  decimals,  is 
2.7182818,  and  is  denoted  by  the  letter  e. 

The  common  system  is  used  in  actual  calculation  ;  the 
natural  system  is  used  in  the  higher  mathematics. 

Exercise  XXIII. 

1.  Given Iogio2=0.30103,logio3=0.47712,  logio 7 =0.84510 
find 

logio6,    logiol4,    logio21,    logio4,    logiol2, 

logio5,    logioi,      logioi,      logiol,    logiofj. 

2.  With  the  data  of  example  1,  find 

logalO,    logaS,    loggS,    log^i,    logs^l^. 


120  TRIGONOMETRY. 

3.  Given  logio  e  =  0.43429  find 

log,2,   log,3,   log,5,   logj,     logeS, 

l0g,9,     log.l,     l0g4,     logelf,     loge/^. 

4.  Find  x  from  the  equations 

5^  =  12,     16^=:  10,     27^  =  4. 

§  43.   Exponential  and  Logarithmic  Series. 
Exponential  Series.    By  the  binomial  theorem 


(-=)■ 


.    ,  1   ,  nx  (nx  —  1)  ^  ^  1 

n  1-2  n^ 


nx  (nx  —  1)  (nx  —  2)       1  ^^ 


1-2-3 


'     /        1\          /        IV        2\ 
x[  X X\  X )[  X 

=  1  +  ^'  + [2 + g + •        (1) 

This  equation  is  true  for  all  real  values  of  x,  since  the 
binomial  theorem  may  readily  be  extended  to  the  case  of 
incommensurable  exponents  (College  Algebra,  §  264);  it  is, 
however,  true  only  for  values  of  ?i  numerically  greater  than 

1,  since  -  must  be  numerically  less  than  1  (College  Algebra, 

§  375). 

As  (1)  is  true  for  all  values  of  x,  it  is  true  when  x  =  1. 


n       \        ^/  \         ^/ 


1  +  1  +  -^+^ f -  + •    (2) 


■■(-=)' 

-  [(-r.)TK-9' 


CONSTRUCTION    OF    TABLES. 

Hence,  from  (1)  and  (2), 
1 


121 


1  +  14 


\2      '  \3 

x[  X X\  X X 


l  +  o: 


12 


^ 


This  last  equation  is  true  for  all  values  of  n  numerically 
greater  than  1.  Taking  the  limits  of  the  two  members  as  n 
increases  without  limit  we  obtain 


( 


1+1+1+1+ 


)x  ^2         ^3 

=l+x+|+|+ 


(3) 


and  this  is  true  for  all  values  of  x.  It  is  easily  seen  that  both 
series  are  convergent  for  all  values  of  x. 

The  sum  of  the  infinite  series  in  parenthesis  is  the  natural 
base  e. 

Hence  by  (3), 

^2  _3 

(4) 


To  calculate  the 

value  0 

1  +  ^ 
f  e  we 

2 
3 
4 
5 
6 
7 
8 
9 

x^      x^ 
0^^       

^|2^|3^ 

proceed  as  follows 
1.000000 

1.000000 

0.500000 

0.166667 

0.041667 

0.008333 

0.001388 

0.000198 

0.000025 

Adding, 

To  ten  places, 

e  = 
e  = 

0.000003 
=  2.71828. 
::  2.7182818284, 

122  TRIGONOMETRY. 

Limit  of  f  1  +  -  J  .     By  the  binomial  theorem, 
I  n(n-l)(n-2)  ^^x' 


1-2-3 


This  equation  is  true  for  all  values  of  n  greater  than  x 
(College  Algebra,  §  375).  Take  the  limit  as  n  increases  with- 
out limit,  X  remaining  finite ;  then 

limit        /i    I  A"_i    ,       1^'.^', 
nmMii^y-^n)  ~    "^ "" "^  [2  "*" [3  "^ 

limit         f  1  _i_  1  1  ^ 

~  n  infinite  \         n)    '  ^^^ 

Logarithmic  Series. 

then  i+^  =  ^.^limit        A    ,  lA" 

n  mfinite  V  nj 

If  n  is  merely  a  large  number,  but  not  infinite, 


( 


l  +  f)=l+cc  +  c, 


where  e  is  a  variable  number  which  approaches  the  limit  0, 
when  n  increases  without  limit.     Hence 


y  — 


y  =  n\ll-{-x-\-e  —  n. 


CONSTRUCTION    OF    TABLES.  123 

If  now  n  becomes  oo,  and  consequently  c  becomes  0,  we 
have 


limit 

^      n  infinite 


n\ll  -\-x  —  n  \ 


Assuming  that  x  is  less  than  1,  we  may  expand  the  right- 
hand  member  of  this  equation  by  the  binomial  theorem.  The 
result  is 

=:Ti.ehG-O^G-OG-^)l+ ] 

2^  [3         li 

/v*^  /y»"  /y»4 

.-.  log,(l  +  a?)  =  a3  — -  +  -  — j  + 

This  series  is  known  as  the  logarithmic  series.  It  is  con- 
vergent only  if  X  lies  between  —  1  and  +1,  or  is  equal  to  +1. 
Even  within  these  limits  it  converges  rather  slowly,  and  for 
these  reasons  it  is  not  well  adapted  to  the  computation 
of  logarithms.  A  more  convenient  series  is  obtained  as 
follows. 

Calculation  of  Logarithms.     The  equation 

log.(H-2/)  =  2/-f  +  f-f  + (1) 

holds  true  for  all  values  of  y  numerically  less  than  1 ;  there- 
fore, if  it  holds  true  for  any  particular  value  of  y  less  than  1, 
it  will  hold  true  when  we  put  —  ?/  for  ?/ ;  this  gives 

iog.(i-y)=-2/-f-f-J- (2) 


124  TRIGONOMETRY. 

Subtracting  (2)  from  (1),  since 

log.  (1  +  y)-  log.  (1  -  2/) = log.  ([±|) , 

wefind    log.(i±f)  =  2(.+  f  +  f  + )• 

Put      ,=  ^;     then^  =  -^. 

and  log,  f  ^-^  J  =  loge  (^  + 1)  -  logc« 

^o?    1     I       1       I       1 Y 

V2^  +  1^3(2^  +  l)3^5(2^  +  l)«^       y 
This  series  is  convergent  for  all  positive  values  of  z. 
Logarithms  to  any  base  a  can  be  calculated  by  the  series  : 
loga(^  +  l)  — log„« 

'^log,aV2;^4-l"*"3(2^  +  l)3"'"5(2^  +  iy"^ /     ^  ^^' 

Calculate  log, 2  to  five  places  of  decimals. 

Let  z=\;        then  z+ 1  =  2,        2z+l  =  3, 

A  1       o      2   ,        2        ,        2        ,        2        . 

and  iog,2  =  5  +  r— — +  — —  + 


3      3  X  33  ■  5  X  35      7  X  37 
The  work  may  be  arranged  as  follows : 


2.000000 


0.666667 -^    1=0.666667 


0.074074 -f-   3  =  0.024691 


0.008230 -f    5  =  0.001646 


0.000914 -^    7  =  0.000131 


0.000102 -f    9  =  0.000011 


0.000011^11  =  0.000001 


loge2  =  0.693147 

Note,  In  calculating  logarithms  the  accuracy  of  the  work  may  be 
tested  every  time  we  come  to  a  composite  number  by  adding  together  the 
logarithms  of  the  several  factors.  In  fact,  the  logarithms  of  composite 
numbers  are  best  found  in  this  way,  and  only  the  logarithms  of  prime 
numbers  need  be  computed  by  the  series. 


CONSTRUCTION    OF    TABLES. 


125 


Exercise  XXIY. 

1.  Calculate  to  five  places  of  decimals  logg3,  logg5,  log^T. 

2.  Calculate  to  ten  places  of  decimals  log^lO. 

3.  Calculate  to  five  places  of  decimals  logio2,  logioe,  logioll. 

§  44.    Trigonometric  Functions  of  Small  Angles. 

Let  A  OF  be  any  angle  less  than  90°  and  x  its  circular 
measure.  Describe  a  circle 
of  unit  radius  about  0  as 
a  centre  and  take  Z.AOP' 
=  —  AA  OP.  Draw  the 
tangents  to  the  circle  at  P 
and  P\  meeting  OA  in  T. 
Then  from  Geometry 
chord  PP'<  arc  PP' 

<PT-{-P^T, 
or,  dividing  by  2 

MP<SiTGAP<PT, 
or  sin  x<Cx<.  tan  x. 

Hence,  dividing  by  sin  x 

X 


Fig,  36. 


or 


1< 
1> 


smic 
since 

X 


<  sec  X, 
>  cos  X. 


(1) 


Then 


since 


lies  between  cos  x  and  1.     If  now  the  angle  x  is 


constantly  diminished,  cos  x  approaches  the  value  1. 


sma? 


Accordingly,  the  limit  of  ,  as  x  approaches  0,  is  1 ; 


since 


or,  in  other  words,  if  cc  is  a  very  small  angle differs  from 

X 

1  by  a  small  value  c,  which  approaches  0  as  cc  approaches  0. 


126  TKIGONOMETRY. 

To  find  the  sine  and  cosine  of  V. 
If  X  is  the  circular  measure  of  Y, 

*  =  ad^O  =  411^  =  0-00029088+, 
the  next  figure  in  x  being  either  7  or  8. 

Now  sina:>  0  but  <«  ;  hence  sin  V  lies  between  0  and  0.000290889. 


Again  cos  1'  =  Vl  —  sin2  V 

>Vl- (0.0003)2 

>    0.9999999. 
Hence  cos  V  -  0. 9999999+. 

But,  from  (1),  sin  ic  >  ic  cos  x 

.-.  sin  l'>  0.000290887  X  0.9999999 

>  0.000290887  (1  —  0.0000001) 

>  0.000290887  —  0.0000000000290887 

>  0.000290886. 

Hence  sinl'  lies  between  0.000290886  and  0.000290889; 
that  is,  to  eight  places  of  decimals 

sin  1'  =  0.00029088 +  , 
the  next  figure  being  6,  7,  or  8. 

Exercise  XXV. 
Given  7r  =  3.141592653589, 

1.  Compute  sin  1',  cos  V,  and  tan  1'  to  eleven  places  of 
decimals. 

2.  Compute  sin  2'  by  the  same  method,  and  also  by  the 
formula  sin  2  a;  =  2  sin  x  cos  x.  Carry  the  operations  to  nine 
places  of  decimals.     Do  the  two  results  agree  ? 

3.  Compute  sin  1°  to  four  places  of  decimals. 

4.  From  the  formula  cos  a;  =  1  —  2  sin^  -,  show  that 
cosa;>  1 — — • 

Li 


CONSTRUCTION    OF    TABLES.  127 

5.  Show  by  aid  of  a  table  of  natural  sines  that  sin  x  and  x 
agree  to  four  places  of  decimals  for  all  angles  less  than  4°  40'. 

6.  If  the  values  of  log  x  and  log  sin  x  agree  to  five  decimal 
places,  find  from  a  table  the  greatest  value  x  can  have. 

§  45.    Simpson's  Method  of  Constructing  a  Trigono- 
metric Table. 

By  §  31  (Plane  Trigonometry)  we  have* 

sin  (^-f^)  +  sin  (^  —  ^)  =2  sin  A  cos  B. 

If  we  put 

A  =  x-\-'^ij,  B  =  ^j, 
this  becomes 

sin  {x-\-^y)-\r  sin  (cc  +  ?/)  =  2  sin  (x-\-2y)  cos y, 
or  sin  (x  +  3?/)  =  2  sin  (ic  +  2?/)  cos  y  —  sin  (x  +  y). 

Similarly  cos  (ic  +  3  y)  =  2  cos  (x-\r2y')  cos  y  —  cos  (x-\-y).  (1) 
li  y  =  V,  the  last  two  equations  become 

sin  (x  +  3')  =  2  sin  (x  +  2')  cos  1'  —  sin  {x  + 1% 
cos  (x  +  3')  =  2  cos  (x  +  2')  cos  1'  —  cos  \x-\-V). 

Hence,  taking  x  successively  equal  to  —  V,  0',  1',  2\ we 

obtain 

sin  2' =  2  sin  1' cos  1', 

sin3'  =  2sin2'cosl'  — sinl', 

sin  4'  =  2  sin  3'  cos  1'  —  sin  2', 


cos2'  =  2cosn'  — 1, 

cos  3'  =  2  cos  2'  cos  1'  —  cos  1', 

cos  4'  =  2  cos  3'  cos  1'  —  cos  2', 


Since  the  sinl'  and  cosl'  are  known,  these  equations  enable 
us  to  compute  step  by  step  the  sine  and  cosine  of  any  angle. 
The  tangent  may  then  be  found  in  each  case  as  the  quotient 
of  the  sine  divided  by  the  cosine. 


128  TRIGONOMETRY. 

This  process  need  be  carried  only  as  far  as  30°.     For 
sin  (30°  +  cc)  +  sin  (30°  —  x)=2  sin 30°  cos  x  =  cos  x, 
cos  (30°  -i-x)  —  cos  (30°  —  4  =  —  2  sin  30°  sin  ic  =  —  sin  x, 
.  • .  sin  (30°  +  ic)  =  cos  X  —  sin  (30°  ~  x), 
cos  (30°  -\-x)==  —  sin  x  +  cos  (30°  —  x). 

Moreover  the  sines  and  cosines  need  be  calculated  only  to 
45°,  since 

sin  (45°  +  x)  =  cos  (45°  —  x), 
cos  (45°  -{-x)  =  sin  (45°  -  x). 
In  using  this  method  the  multiplication  by  cos  1',  which 
occurs  at  each  step,  can  be  simplified  by  noting  that 
cos  V  =  0.9999999  =  1  -  0.0000001. 

Simpson's  method  is  superseded  in  actual  practice  by  much  more  rapid 
and  convenient  processes  in  which  we  employ  the  expansions  of  the 
trigonometric  functions  in  infinite  series. 

Exercise  XXVI. 

1.  Compute  the  sine  and  cosine  of  6'  to  seven  decimal  places. 

2.  In  the  formula  (1)  let  2/=:r.  Assuming  sin  1°=0.017454  +  , 
cos  1°= 0.999848  +  ,  compute  the  sines  and  cosines  from  degree 
to  degree  as  far  as  4°. 

§  46.    De  Moivre's  Theorem. 
Expressions  of  the  form 

cos  x-^i  sin  x, 
when  i  =  V —  1,  play  an  important  part  in  modern  analysis. 
Given  two  such  expressions 

cos  x-\-i  sin  x,   cos  y-\-i  sin  y, 
their  product  is 

(cos  x-{-i  sin  x)  (cos  y-\-i  sin  y) 

=  cos  X  COS  y  —  sin  x  sin  7/-{-i  (cos  x  sin  y  -}-  sin  x  cos  y) 
=  cos  (x-\-y')-\-  i  sin  (x-\-y). 


CONSTRUCTION    OF    TABLES.  129 

Hence,  the  product  of  two  expressions  of  the  form  cos  x 
-\-  i  sin  X,  cos  y-\-i  sin  y  is  an  expression  of  the  same  form  in 
which  a;  or  ?/  is  replaced  by  x-{-y.  In  other  words,  the  angle 
which  enters  into  such  a  product  is  the  sum  of  the  angles  of 
the  factors. 

If  X  and  y  are  equal,  we  have  at  once  from  the  preceding 

(cos  x-]ri  sin  xy  =  cos  2x-\-ism2x] 
and  again 

(cos  x-\-i  sin  x)^  =  (cos  x-\-i  sin  xy  (cos  x-^i  sin  x) 

=  (cos 2x-{-ism2x)  (cos x-\-i sin x) 

=  cos  3  a?  4"  ^  sin  3  cc. 

Similarly         (cos  x-{-i  sin  xy  =  cos  4:X-{-i  sin  4a;, 
and  in  general  if  ?^  is  a  positive  integer 

(cos  x-\-  i  sin  a;)"  =  cos  nx  -\-  i  sin  tix.  (1) 

Hence 

To  raise  the  expression  cos  x -\- i  sin  x  to  the  nth  power  when 
n  is  a  positive  integer,  we  have  only  to  multiply  the  angle  x  by  n. 

Again,  if  ^  is  a  positive  integer  as  before, 

/       X  .   .  .    x\  ... 

I  cos-  +  iSin-  )  =  cos  a;  +  * sm  a; 
\       n  nj 

.*.  (cos  a;  +  i  sin  aj>  =  cos  -  +  ^  sin — 

n  n 

Since,  however,  x  may  be  increased  by  any  integral  multiple 
of  2  TT  without  changing  cos  x-\-i  sin  x,  it  follows  that  all  the 
n  expressions 

x  ,    .  .    x  a  +  27r  ,    .   .    x-\-2Tr 

cos-  +  isin->       cos f-^sln j 

n  n  n  n 

x-\-4:'ir  .    .    .    x-\-4:'jr 
cos h  t  sm } ) 


cos 


x-\-(n  —  l)27r  ,    .    .    x-\-(n  —  l)2'jr 
— ^-^ J-  ^  sm ^ 


130  TRIGONOMETRY. 

are  nth.  roots  of  cos  x-j-i  sin  x.     There  are  no  other  roots,  since 

x-\-n27r  ,    .   .    x-\-n2'7r 
cos \- 1  sm 


X 

isin-7 
n 


=  cos  (  -  +  27r  )  +isin  (  -  +  27r  )  —  cos-  + 
\n         J  y^         J  ^ 

and  cos ^ h  *  sm ^ ^ — 

=  cos  (   — ' [-27r  1  4-^Sln  (  — ' K^tt  1 

=  cos h  t  sm > 

n  n 

and  so  on. 

Hence,  if  w  is  a  positive  integer, 
1 
(cos  x-\-i  sin  ic)« 

=  cos — ' f-*sm— ! (k  — 0,1,2, n  —  l).    (2) 

From  (1)  and  (2)  it  follows  at  once  that  if  m  and  n  are 
positive  integers 

(cos  x-\-{^mxY  =^\  (cos  £c  +  /  sin  a?)«  !- 
=cos-(a;+2A:7r)+*'sin  — (£c-l-27c7r)(A;=0,l,2, w— 1).  (3) 

Finally,  if is  a  negative  fraction, 

71) 

TO  1 

(cos  £c  +  ^  sin  a?)    « 


But 


(cosa;-}-^' since)" 

1 cos  X  —  {  sin  X 

cos  x-\-i  sin  x      (cos  x-\-i  sin  £c)  (cos  a;  —  i  sin  a;) 

cosic  —  -isinx 

cos^  a?  +  i  sin^  a; 
=  cos  X  —  i  sin  x , 
=  cos  ( —  a?)  +  ^  sin  (—  x). 


CONSTRUCTION    OF    TABLES.  131 

Hence 

(cos  x-\-i  sin  x)    «  ^  k  cos  ( —  x)  +  ^  sin  (—  x)  p 

771  TTl 

=  cos  —  (— cc  +  2A;7r)  +  *'sin  — (— a:  +  2A;7r), 

(k  =  0,  1,  2, n  —  1) 

'■  (x-{- 2kir)  ^  +  *' sin  -^ {x-\-2  Jctt)  j> , 

(k  =  0,l,2, 71-1).        (4) 

Consequently  if  ti  is  a  positive  or  negative  integer  or  fraction 

(cos  x-jri sin  af)"  =  cos  [n(x-\^2  kir)^  -{-  i  sin  [w  (cc  +  2  A^tt)], 

(7c  =  0,  1,  2, n-1).        (5) 

Example  :  Find  the  three  cube  roots  of  —  1. 

We  have  —  1  =  cos  180°  +  i  sin  180° 

,     ,,,             180°  +  2A:7r  ,    .   .    180°  +  2A;7r,,       ^  ,   „, 
.  •.  (—  1)^  =  cos +  I  sm {k  =  0, 1,  2). 

For  the  three  cube  roots  of  —  1  we  find  therefore 

cos  60°  +  i  sin  60°,        cos  180°  +  i  sin  180°,        cos  300°  +  i  sin  300°, 

l  +  iVs  ,  l-iVs 

-^— '  -1'  -^- 

By  aid  of  De  Moivre's  Theorem  we  may  express  sin  716  and 

cos  nO,  when  n  is  an  integer,  in  terms  of  sin  0  and  cos  6. 

Thus 

cos  n6  +  i  sin  tiO  =  (cos  0-\-i  sin  Oy 

71  {71  —  1  I 

=  COS"  0-{-i7i  cos"-^  ^  sin  ^  +  P  -^ — ^  cos"-^  0  sin^  0 

_^.3^(n-l)(.-2)^^^^_3^^.^3^^  

Or,  sincei^=  — 1,  i^=  —  i,  {*=-}-l,  

cos  w  ^  4"  *'  sin  w  ^  =  cos"  6-{-i7i  cos"~^  ^  sin  ^ 

2  |o 


132  TRIGONOMETRY. 


Equating   now   the   real   parts   and   the   imaginary   parts 
separately,  we  obtain 

2 


Gosne  =  cos"  0  —  '^^^ — ^  cos"-2  0  sin^  0 


,  n(n-l)(n-2)(n-S)           .,   .   ,. 
+  — ^^- — ^-^^ — ^ — ^cos**-^^sin*^— 

sin  n$  =  n  cos"~^  ^  sin  ^ ^^^ — r^ cos""^  6  sin^  6 

H ^^ ^^ r~ — ^  cos**-^  ^  sm^  ^  — 

Exercise  XXVII. 

1.  Find  the  six  6th  roots  of  —  1 ;  of  + 1. 

2.  Find  the  three  cube  roots  of  i. 

3.  Find  the  four  4th  roots  of  —  i. 

4.  Express  sin  4  0  and  cos  4  ^  in  terms  of  sin  0  and  cos  0. 

§  47.   Expansion  of  Sin  x,  Cos  x,  and  Tan  x  in 
Infinite  Series. 

Let  one  radian  be  denoted  simply  by  1,  and  let 

cos  1  +  i  sin  l  =  k. 
Then  cos  x-\-i  sin  x  =  (cos  1  +  i  sin  1)^  =  k^, 

and  putting  — x  for  x 

cos  ( —  x)-\-i  sin  ( —  x)  =  cos  x  —  i  sin  x  =  k~^. 

That  is 

cosa;4-isina;=  A:^ 
and  cos  x  —  i  sin  x  =  k~^ 

By  taking  the  sum  and  difference  of  these  two  equations, 
and  dividing  the  sum  by  2  and  the  difference  by  2  i,  we  have 

cos  a?  =  ^  (7c^  +  k-^),         sin  x  =  ~  (k"^  —  k-^). 
But  k^  =  (e  '^^  ^')  *  =  e^  ^^  *,        k'""  =  e"^  ^^  *, 


CONSTRUCTION    OF    TABLES.  133 

and 

2i  \o 

1    /7r     .      7      rX  -.       .     ^^^  (log  A;)^     ,     iC*  (log  A^V     , 

.-.  cosx  =  i(A:^  +  A:-^)  =  l  +  — ^^-^  +  — ^T^^  + 


sinx  =  -^  a: log  a; -|-        ,0         r       .^ 


It  only  remains  to  find  the  value  of  ^,  and  this  can  be 
obtained  by  dividing  the  last  equation  through  by  x  and 
letting  X  approach  0  indefinitely,  when  we  have 

limit 

X 


limit      /sin  icX       1 . 

.  • .  log  li  =  i,         k^  e\ 
Therefore  we  have 


X  X  X 

COS  a;  =  i  (e-+  e— ■)  :=!_-  + _ --4- 

From  the  last  two  series  we  obtain  by  division 

since  ,   x^  ,  2x^  ,   11  x'  , 

cosa:  0        15        olo 

By  the  aid  of  these  series  the  trigonometric  functions  of 
any  angle  are  readily  calculated.  In  the  computation  it 
must  be  remembered  that  x  is  the  circular  measure  of  the 
given  angle. 


134  TRIGONOMETRY. 

Exercise  XXVIII. 
Verify  by  the  series  just  obtained  that 

1.  sin^ic  +  cos^x  =  1. 

2.  sin(— cc)  =  —  sinic,  cos  (— x)  =  cos  a;. 

3.  siii2£c  =  2siiia;cosic.  4.    cos  2  ic  =  1  —  2  sin^ic. 

5.  Find  the  series  for  sec  x  as  far  as  the  term  containing 
the  6th  power  of  x. 

X 

6.  Find  the  series  for  x  cot  x,  noting  that  a;  cot  ic  =  — —  cos  x. 

7.  Calculate  sin  10°  and  cos  10°  to  6  places  of  decimals. 

8.  Calculate  tan  15°  to  5  places  of  decimals. 

From  the  exponential  values  of  sin  x  and  cos  x  show  that 

9.  cos3cc  =  4cos^a;  —  3coscc. 
10.   sin  3  X  =  3  sin  ic  —  4  sin^x. 


SPHEEICAL  TEIGONOMETEY. 


CHAPTER   VII. 

THE   RIGHT    SPHERICAL    TRIANGLE. 
§  48.   Introduction. 

The  object  of  Spherical  Trigonometry  is  to  show  how 
spherical  triangles  are  solved.  To  solve  a  spherical  triangle 
is  to  compute  any  three  of  its  parts  when  the  other  three  parts 
are  given. 

The  sides  of  a  spherical  triangle  are  arcs  of  great  circles. 
They  are  measured  in  degrees,  minutes,  and  seconds,  and 
therefore  by  the  plane  angles  formed  by  radii  of  the  sphere 
drawn  to  the  vertices  of  the  triangle.  Hence,  their  measures 
are  independent  of  the  length  of  the  radius,  which  may  be 
assumed  to  have  any  convenient  numerical  value;  as,  for 
example,  unity. 

The  angles  of  the  triangle  are  measured  by  the  angles  made 
by  the  planes  of  the  sides.  Each  angle  is  also  measured  by 
the  number  of  degrees  in  the  arc  of  a  great  circle,  described 
from  the  vertex  of  the  angle  as  a  pole,  and  included  between 
its  sides. 

The  sides  may  have  any  values  from  0°  to  360°;  but  in  this 
work  only  sides  that  are  less  than  180°  will  be  considered. 
The  angles  may  have  any  values  from  0°  to  180°. 

If  any  two  parts  of  a  spherical  triangle  are  either  both  less 
than  90°  or  both  greater  than  90°,  they  are  said  to  be  alike  in 
kind;  but  if  one  part  is  less  than  90°,  and  the  other  part 
greater  than  90°,  they  are  said  to  be  unlike  in  kind. 


136 


SPHERICAL    TRIGONOMETRY. 


Spherical  triangles  are  said  to  be  isosceles,  equilateral, 
equiangular,  right,  and  oblique,  under  the  same  conditions  as 
plane  triangles.  A  right  spherical  triangle,  however,  may 
have  one,  two,  or  three  right  angles. 

When  a  spherical  triangle  has  one  or  more  of  its  sides  equal 
to  a  quadrant,  it  is  called  a  quadrantal  triangle. 

It  is  shown  in  Solid  Geometry,  that  in  every  spherical 
triangle 

I.  If  two  sides  of  a  spherical  triangle  are  unequal,    the 
angles   opposite   them   are  unequal,   and  the  greater  angle  is 
opposite  the  greater  side;  and  conversely. 
II.   The  sum  of  the  sides  is  less  than  360°. 

III.  The  sum  of  the  angles  is  greater  than  180°  and  less 
than  540°. 

IV.  If,  from  the  vertices  as  poles,  arcs  of  great  circles  ai'e 
described,  another  spherical  triangle  is  formed  so  related  to  the 
first  triangle  that  the  sides  of  each  triangle  are  supplements  of 
the  angles  opposite  them  in  the  other  triangle. 

Two  such  triangles  are  called  ^oZar  triangles,  or  supplemental 
triangles. 

Let  A,  B,   C  (Fig.  37)  denote  the  angles  of  one  triangle ; 

a,  h,  c  the  sides  opposite  these 
angles  respectively ;  and  let  A\ 
B',  C  and  a',  b',  c'  denote  the 
corresponding  angles  and  sides 
of  the  polar  triangle.  Then  the 
above  theorem  gives  the  six 
following  equations : 

^  +  ^'  =  180°, 
^+^^'  =  180°, 
C+^'==180°, 
^'  +  ^=180°, 
B'-^b  =  180°, 
C'+c=  180°. 


THE    RIGHT    SPHERICAL    TRIANGLE.  137 

Exercise  XXIX. 

1.  The  angles  of  a  triangle  are  70°,  80°,  and  100°;  find  the 
sides  of  the  polar  triangle. 

2.  The  sides  of  a  triangle  are  40°,  90°,  and  125°;  find  the 
angles  of  the  polar  triangle. 

3.  Prove  that  the  polar  of  a  quadrantal  triangle  is  a  right 
triangle. 

4.  Prove  that,  if  a  triangle  has  three  right  angles,  the  sides 
of  the  triangle  are  quadrants. 

5.  Prove  that,  if  a  triangle  has  two  right  angles,  the  sides 
opposite  these  angles  are  quadrants,  and  the  third  angle  is 
measured  by  the  number  of  degrees  in  the  opposite  side. 

6.  How  can  the  sides  of  a  spherical  triangle,  given  in 
degrees,  be  found  in  units  of  length,  when  the  length  of  the 
radius  of  the  sphere  is  known? 

7.  Find  the  lengths  of  the  sides  of  the  triangle  in  Example  2, 
if  the  radius  of  the  sphere  is  4  feet. 

§  49.   Formulas  Kelating  to  Right  Spherical  Triangles. 

As  is  evident  from  §  48,  Examples  4  and  5,  the  only  kind 
of  right  spherical  triangle  requiring  further  investigation  is 
that  which  contains  07ily  one  right  angle. 

Let  ABC  (Fig.  38)  be  a  right  spherical  triangle  having 
only  one  right  angle;   and  let  A,  B,  C 
denote  the  angles  of  the  triangle;  a,  b,  c, 
respectively,  the  opposite  sides. 

Let  CJoeJih^jdght  angle ;  and  for  the 
present  suppose  that  each  of  the  other 
parts  is  less  than  90°,  and  that  the  radius 
of  the  sphere  is  1. 

Let  planes  be  passed  through  the  sides, 
intersecting  in  the  radii  OA,  OB,  and  OC, 


138 


SPHERICAL    TRIGONOMETRY. 


Also,  let  a  plane  perpendicular  to  OA  be  passed  through  B, 
cutting  OA  at  ^  and  OC  at  D.  Draw  BE,  BD,  and  BE. 
BE  and  BE  are  each  J_  to  OA  (Geom.  §  462) ;  therefore 
Z  BEB  =  A.  The  plane  BBE  is  _L  to 
the  plane  AOC  (Geom.  §  518);  hence  BB, 
which  is  the  intersection  of  the  planes 
BBE  and  BOC,  is  ±  to  the  plane  AOC 
(Geom.  §  520),  therefore  _L  to  OC  and 
BE. 

Now     cos  c  =  0E=  OB  X  cos  b, 
and  OB  =  cos  a. 

cos  c  =  COS  a  cos  b. 
sin  a  =  BB  —  BE  X  sin  A, 
and  BE  =  sin  c. 

sin  a  =  sin  c  sin  A 
sin  b  =  sin  c  sin  B 

BE      OE  tsiiib 


Fig.  39. 
Therefore, 

Therefore, 
changing  letters. 


} 


[38] 


[39] 


Hence, 
changing  letters. 


""""^^-BE-OEt'^nG 
cos  A  =  tan  b  cot  c 
cos  B  ==  tan  a  cot  c . 


} 


[40] 


Again, 


cos^ 


BE       OB  sin  b 


By  substituting  for 

changing  letters. 

Also, 

Hence, 
changing  letters, 


BE 

sinb 


sine 


=  cos  a 


sin  b 


sm  c 


its  value  from  [39],  we  obtain 

}  [41] 


sm  c 

COS  A  =  COS  a  sin  B 
cos  B  =  COS  b  sin  A 


sin  b  =  -r—  = 


BE      BB  cot  A 


OB  OB 

sin  b  =  tan  a  cot  A 
sin  a  =  tan  b  cot  B 


=  tan  a  cot  A. 


} 


[42] 


If  in  [38]  we  substitute  for  cos  a  and  cos  b  their  values  from 

[41],  we  obtain 

cos  c  =  cot  A  cot  B.  [43] 

Note.    In  order  to  deduce  the  second  formulas  in  [39]-[42]  geometri- 
cally, the  auxiliary  plane  must  be  passed  through  -4  _L  to  OB. 


THE    RIGHT    SPHERICAL    TRIANGLE. 


139 


These  ten  formulas  are  sufficient  for  the  solution  of  any 
right  spherical  triangle. 

In  deducing  these  formulas,  it  has  been  assumed  that  all 
the  parts  of  the  triangle,  except  the  right  angle,  are  less 
than  90°.  But  the  formulas  also  hold  true  when  this  hypoth- 
esis is  not  fulfilled. 

Let  one  of  the  legs  a  be  greater  than  90°,  and  construct  a 
figure  for  this  case  (Fig.  40)  in  the  same  manner  as  Fig.  38. 


Fig.  40.     , 

The  auxiliary  plane  BDE  will  now  cut  both  CO  and  AO  pro- 
duced beyond  the  centre  0 ;  and  we  have 

gosc  =  —OjE=—OD  cos  DOE 

=  (—  cos  a)  (—  cos  b) 
=  cos  a  cos  b. 
Likewise,  the  other  formulas,  [39]-[43],  hold  true  in  this  case. 
Again,  suppose  that  both  the  legs  a  and  b  are  greater  than 
90°.     In  this  case  the  plane  BDE  (Fig.  41)  will  cut  CO  pro- 
duced beyond  0,  and  A  0  between  A  and  0 ;  and  we  have 
cos  c=OE  =  OD  cos  DOE 

=  (—  cos  a)  (—  cos  b) 
=  cos  a  cos  b, 

a  result  agreeing  with  [38].      And  the  remaining  formulas 
may  be  easily  shown  to  hold  true. 

Like  results  follow  in  all  cases ;   in  other  words,  Formulas 
[38]-[43]  are  universally  true. 


140  SPHERICAL    TRIGONOMETRY. 

Exercise  XXX. 

1.  Prove,  by  aid  of  Formula  [38],  that  the  hypotenuse  of 
a  right  spherical  triangle  is  less  than  or  greater  than  90°, 
according  as  the  two  legs  are  alike  or  unlike  in  kind. 

2.  Prove,  by  aid  of  Formula  [41],  that  in  a  right  spherical 
triangle  each  leg  and  the  opposite  angle  are  always  alike  in 
kind. 

3.  What  inferences  may  be  drawn  from  Formulas  [38]-[43] 
respecting  the  values  of  the  other  parts:  (i.)  if  c  =  90°; 
(ii.)  iia  =  90°  ;  (iii.)  if  c  =  90°  and  a  =  90°  ;  (iv.)  \i  a  =  90° 
and  ft  =  90°? 

Deduce  from  [38] -[43]  and  [18] -[23]  the  following 
formulas  : 

4.  tan2^&  =  tan^(c  — a)  tan|-(c4-^). 

Hint.  Use  Formula  [18]  and  substitute  in  it  the  value  of  cos  6  in 
[38]. 

5.  tan^  (45°  —  I  ^)  =  tan  i  (c  —  a)  cot  ^  (c  +  tt) . 

6.  tan^ iB  =  sin (c  —  a) esc (c -f- <x). 

7.  tan2^c  =  — cos(^+^)sec(^— ^). 

8.  tan^  ^a  =  tan  [|  (A+B)  —  45°]  tan  [-J  (A—B)-\-  45°]. 

9.  tan^  (45°  —  |-c)  =  tan  4-  (vl  —  a)  cot  i(A  +  a). 

10.  tan2  (45°  -ib)  =  sin  (A  —  a)  esc  (A  -f  a). 

11.  tan2 (45°  -iB)  =  tan  i(A-a)  tan  i(A-}-a). 


THE    RIGHT    SPHERICAL    TRIANGLE. 


141 


§  50.   Napier's  Eules. 

The  ten  formulas  deduced  in  §  49  express  the  relations 
between  five  parts  of  a  right  triangle,  the  three  sides  and 
the  two  oblique  angles.  All  these  relations  may  be  shown  to 
follow  from  two  very  useful  Rules,  devised  by  Baron  Napier, 
the  inventor  of  logarithms. 

For  this  purpose  the  right  angle  (not  entering  the  formulas) 
is  left  out  of  account,  and  instead  of  the  hypotenuse  and  the 
two  oblique  angles,  their  respective  complements  are  employed; 
so  that  the  five  parts  considered  by  the  Eules  are :  a,  h,  go.  A, 
CO.  c,  CO.  B.  Any  one  of  these  parts  may  be  called  a  middle 
part ;  and  then  the  two  parts  immediately  adjacent  are  called 
adjacent  parts,  and  the  other  two  are  called  opposite  parts. 

Eule  I.  The  sine  of  the  middle  part  is  equal  to  the  product 
of  the  tangents  of  the  Adjacent  parts. 

Eule  II.  The  sine  of  the  middle  part  is  equal  to  the  product 
of  the  cosines  of  the  opposite  parts. 

These  Eules  are  easily  remembered  by  the  expressions, 
tan.  ad.  and  cos.  op. 

The  correctness  of  these  Eules  may  be  shown  by  taking 
each  of  the  five  parts  as  middle  part,  and  com- 
paring the  resulting  equations  with  the  equa- 
tions contained  in  Formulas  [38]-[43]. 

For  example,  let  co.  c  be  taken  as  middle 
part,  then  co.  A  and  co.  B  are  the  adjacent  parts, 
and  a  and  5  the  opposite  parts,  as  is  very  plainly 
seen  in  Fig.  42.     Then,  by  Napier's  Eules : 

sin  (co.  c)  =  tan  (co.  A)  tan  (co.  B), 
or  -cos  c  =  cot  A  cot  B ; 

sin  (co.  c)  =  cos  a  cos  b, 
or  cos  c  =  cos  a  cos  b ; 

results  which  agree  with  Formulas  [43]  and  [38]  respectively. 


142  SPHEKICAL    TRIGONOMETRY. 

Exercise  XXXI. 

1.  Show  that  Napier's  Rules  lead  to  the  equations  con- 
tained in  Formulas  [39],  [40],  [41],  and  [42]. 

2.  What  will  Napier's  Eules  become,  if  we  take  as  the  five 
parts  of  the  triangle,  the  hypotenuse,  the  two  oblique  angles, 
and  the  complements  of  the  two  legs  ? 

§  51.    Solution  of  Eight  Spherical  Triangles. 

By  means  of  Formulas  [38]- [43]  we  can  solve  a  right  tri- 
angle in  all  possible  cases.  In  every  case  two  parts  besides 
the  right  angle  must  be  given. 

Case  I.    Given  the  two  legs  a  and  h. 

The  solution  is  contained  in  Formulas  [38]  and  [42] ;  viz. : 

cos  c  =  cos  a  cos  by 

tan  A  =  tan  a  esc  b, 

tan  B  =  tan  ^  CSC  a. 

For  example,  let  a  =  27°  28'  36",  b  =  51°  12'  8" ;  then  the 
solution  by  logarithms  is  as  follows  : 

log  cos  a  =  9.94802 
log_cos_^^9^79697 
log  cose  =  9.74499 

c  =  56°  13' 41" 


log  tan  a  =9.71605 
log  CSC  b  =0.10826 
log  tan  ^  =  9.82431 

^  =  33°  42' 51" 


log  tan  ^»  =10.09476 
log  CSC  a  =   0.33594 
log  tan  ^  =  10.43070 
^  =  69°  38' 54" 


Case  II.    Given  the  hypotenuse  c  and  the  leg  a. 

From  Formulas  [38],  [39],  and  [40]  we  obtain 
cos  J  =  COS  c  sec  a, 
sin  A  =  sin  a  esc  c, 
cos  B  =  tan  a  cot  c. 


THE    RIGHT    SPHERICAL    TRIANGLE.  143 

Although  two  angles  in  general  correspond  to  sin^,  one 
acute  the  other  obtuse,  yet  in  this  case  the  indetermination 
is  removed  by  the  fact  that  A  and  a  must  be  alike  in  kind 
(see  Exercise  XXX.,  Example  2). 

Case  III.    Given  the  leg  a  and  the  opposite  angle  A. 

By  means  of  Formulas  [39],  [42],  and  [41],  we  find  that 

sin  G  =  sin  a  esc  A, 
sin  b  =  tan  a  cot  A, 
sin  B  =  sec  a  cos  A  ; 

or,  from  [38]  and  [40], 

cos  h  =  cos  c  sec  a, 
cos  B  =  tan  a  cot  c. 

When  c  has  been  computed,  b  and  B  are  determined  by  these 
values  of  their  cosines ;  but,  since  c  must  be  found 
from  its  sine,  c  may  have  in  general  two  values 
which  are  supplements  of  each  other.     This  case, 
therefore,  really  admits  of  two  solutions. 

In  fact,  if  the  sides  b  and  c  are  extended  until 
they  meet  in  A'  (Fig.  43),  the  two  right  triangles 
ABC  and  A'BC  have  the  side  a  in  common,  and 
the  angle  A  =  A'.  Also  A'C  =  1S0°  —  b,  A'B 
=  180°  -  c,  and  ZA'BC  =  180°  -  B. 

Case  IV.    Given  the  leg  a  and  the  adjacent  angle  B. 

Formulas  [40],  [42],  and  [41]  give 

tan  c  =  tan  a  sec  B, 

tan  b  =  sin  a  tan  Bj    •^"■'^^    > 

cos  A  =  cos  a  sin  B. 


144  SPHERICAL    TRIGONOMETRY. 

Case  V.    Given  the  hypotenuse  c  and  the  angle  A. 
From  Formulas  [39],  [40],  and  [43]  it  follows  that 

sin  a  =  sin  c  sin  Aj 

tan  b  =  tan  c  cos  A, 

cot  B  =  cos  c  tan  A. 

Here  a  is  determined  by  sin  a,  since  a  and  A  must  be  alike 
in  kind  (see  Exercise  XXX.,  Example  2). 

Case  VI.    Given  the  two  angles  A  and  B. 

By  means  of  Formulas  [43]  and  [41]  we  obtain 

cos  c  =  cot  ^  cot  B, 

cos  a  =  cos  ^  CSC  B, 

cos  b  =  COS  B  CSC  A. 

Note  1.  In  Case  I.  (a  and  6  given)  the  formula  for  computing  c  fails 
to  give  accurate  results  when  c  is  very  near  0°  or  180°;  in  this  case  it  may- 
be found  with  greater  accuracy  by  first  computing  B,  and  then  computing  c, 
as  in  Case  IV. 

Note  2.  In  Case  II.  (c  and  a  given),  if  6  is  very  near  0°  or  180°,  it 
may  be  computed  more  accurately  by  means  of  the  derived  formula 

tan2  i  6  =  tan  i  (c  +  a)  tan  \  (c  —  a).  (Ex.  4,  §  49. ) 

And  if  A  is  so  near  90°  that  it  cannot  be  found  accurately  in  the  Tables, 
it  may  be  computed  from  the  derived  formula 

tan2  (45°  -  i^)  =  tan  i  (c  -  a)  coti  (c  +  a).  (Ex.  5,  §  49.) 

In  like  manner,  when  B  cannot  be  accurately  found  from  its  cosine  we 
may  make  use  of  the  formula 

tan2 ^  ^  =r  sin  (c  —  a)  esc  (c  +  a).  (Ex.  6,  §  49. ) 

Note  3.  In  Case  III.  (a  and  A  given),  when  the  formulas  for  the 
required  parts  do  not  give  accurate  results,  we  may  employ  the  derived 
formulas 

tan2  (45°  -  i c)  =  tan  i  {A-a)  cot i  (^  +  a),  (Ex.  9,  §  49.) 

tan2  (45°  -  i  6)  =  sin  {A  —  a)  esc  {A  +  a),  (Ex.  10,  §  49.) 

tan2  (45°  -  i  E)  =  tan  i  (^  -  a)  tan  i  (^  +  a).        (Ex.  11,  §  49.) 


THE    RIGHT    SPHERICAL    TRIANGLE.  145 

Note  4.  In  Case  IV.  (a  and  B  given),  if  A  is  near  0°  or  180°,  it  may 
be  more  accurately  found  by  first  computing  b  and  tlien  finding  A. 

Note  5.  In  Case  V.  (c  and  A  given),  if  a  is  near  90°,  it  may  be  found 
by  first  computing  6,  and  then  computing  a  by  means  of  Formula  [42]. 

Note  6.  In  Case  VI.  {A  and  B  given),  for  unfavorable  values  of  the 
sides  greater  accuracy  may  be  obtained  by  means  of  the  derived  formulas 

tan2 ic  =  -cos{A  +  B)  sec  (A  —  B),  (Ex.  7,  §  49.) 

tan2 i a  =  tan  [i  {A  +  B)-  45°]  tan  [45°  +  ^{A-B)'\,  (Ex.  8,  §  49.) 
tan2  ^  6  =  tan  [J  {A  +  B)-  45°]  tan  [45°  -  i(J.  -  B)'\. 

Note  7.  In  Cases  I.,  IV.,  and  V.,  the  solution  is  always  possible. 
In  the  other  Cases,  in  order  that  the  solution  may  be  possible,  it  is 
necessary  and  sufficient  that  in  Case  II.  sin  a  <C  sin  c ;  in  Case  III. ,  that 
a  and  A  be  alike  in  kind,  and  sin  J.  >  sin  a ;  in  Case  VI. ,  that  A+B  +  C 
be  >  180°,  and  the  difference  of  A  and  i?  be  <  90°. 

Note  8.  It  is  easy  to  trace  analogies  between  the  formulas  for  solving 
right  spherical  triangles  and  those  for  solving  right  plane  triangles.  The 
former,  in  fact,  become  identical  with  the  latter  if  we  suppose  the  radius 
of  the  sphere  to  be  infinite  in  length ;  in  which  case  the  cosines  of  the 
sides  become  each  equal  to  1,  and  the  ratios  of  the  sines  of  the  sides  and 
of  the  tangents  of  the  sides  must  be  taken  as  equal  to  the  ratios  of  the 
sides  themselves. 

Note  9.  In  solving  spherical  triangles,  the  algebraic  sign  of  the 
functions  must  receive  careful  attention.  If  the  sign  of  each  function  is 
written  just  above  it,  the  sign  of  the  function  in  the  first  member  will  be 
+  or  —  according  to  the  rule  that  like  signs  give  +  and  unlike  signs 
give  — . 

If  the  function  is  a  cos,  tan,  or  cot,  the  +  sign  shows  that  the  angle  is 
less  than  90°;  the  —  sign  shows  that  the  angle  is  greater  than  90°,  and 
the  supplement  of  the  angle  obtained  from  the  table  must  be  taken. 

If  the  function  is  a  sine,  since  the  sine  of  an  angle  and  its  supplement 
are  the  same,  the  acute  angle  obtained  from  the  table  and  its  supplement 
must  be  considered  as  solutions,  unless  there  are  other  conditions  that 
remove  the  ambiguity.  For  the  conditions  that  remove  the  ambiguity, 
in  case  of  right  spherical  triangles  see  examples  1  and  2  in  Exercise  XXX., 
and  in  case  of  oblique  spherical  triangles  see  I.  of  §  48. 


146 


SPHERICAL    TRIGONOMETRY. 


Note  10.  The  solutions  of  a  spherical  triangle  may  conveniently  be 
tested  by  substituting  them  in  the  formula  containing  the  three  required 
parts. 

If  the  formula  required  for  any  case  is  not  remembered,  it 
is  always  easy  to  find  it  by  means  of  Napier's  Rules.  In 
applying  these  Eules  we  must  choose  for  the  middle  part  that 
one  of  the  three  parts  considered  —  the  two  given  and  the  one 
required  —  which  will  make  the  other  two  either  adjacent 
parts  or  opposite  parts. 

For  example  :  given  a  and  B ;  solve  the  triangle. 

First,  represent  the  parts  as  in  Fig.  42,  and  to  prevent 
mistakes  mark  each  of  the  given  parts  with  a 
cross.  To  find  h,  take  a  as  the  middle  part ; 
then  b  and  co.  B  are  adjacent  parts ;  and  by 
Eule  I., 

sin  a  =  tan  b  cot  B ; 
whence,  tan  b  =  sin  a  tan  B. 

To  find  c,  take  co.  B  as  middle  part ;  then  a 
and  CO.  c  are  adjacent  parts  ;  and  by  Eule  I., 

cos  ^=  tan  a  cot  e\ 
whence,  tan  c  =  tan  a  sec  B. 


To  find  A,  take  co.  A  as  middle  part ; 
and  CO.  B  are  the  opposite  parts  ;  and  by  Eule  II., 

cos  A  ==  cos  a  sin  B. 


then  a 


In  like  manner,  every  case  of  a  right  spherical  triangle  may 
be  solved. 


Exercise  XXXII. 

Solve  the  following  right  triangles,  taking  for  the  given 
parts  in  each  case  those  printed  in  columns  I.  and  II. : 


THE    RIGHT    SPHERICAL    TRIANGLE. 


147 


Note.  The  values  in  the  last  three  columns  of  example  9  cannot  be 
combined  promiscuously  with  those  given  in  columns  I.  and  II. 

Since  a<90°,  with  the  value  of  6>90°  must  be  taken  angle  5>90° 
and  c  >  90° ;  while  with  the  value  of  6  <  90°  must  be  taken,  for  the  same 
reason,  angle   5<90°  and  c<90°.     Exercise  XXX.,  1  and  2. 


148  SPHERICAL    TRIGONOMETRY. 

23.  Define  a  quadrantal  triangle,  and  show  how  its  solution 
may  be  reduced  to  that  of  the  right  triangle. 

24.  Solve  the  quadrantal  triangle  whose  sides  are : 

«=:  174°  12' 49.1",  ^»  =  94°8'20",  c  =  90°. 

25.  Solve  the  quadrantal  triangle  in  which 

G  =  90%  ^  =  110°  47' 50",  ^  =  135°  35' 34.5". 

26.  Given  in  a  spherical  triangle  A,  C,  and  c  each  equal  to 
90°;  solve  the  triangle. 

27.  Given  A  =  60°,  C  =  90°,  and  c  =  90° ;  solve  the  triangle. 

28.  Given  in  a  right  spherical  triangle,  A  =  42°  24'  9", 
^  =  9°  4'  11";  solve  the  triangle. 

29.  In  a  right  spherical  triangle,  given  a  =  119°  11', 
B  =  126°  54';  solve  the  triangle. 

30.  In  a  right  spherical  triangle,  given  c=50°,  ^'==44°18'39"; 
solve  the  triangle. 

31.  In  a  right  spherical  triangle,  given  A  =  156°  20'  30", 
a  =  65°  15' 45";  solve  the  triangle. 

32.  If  the  legs  a  and  6  of  a  right  spherical  triangle  are 
equal,  prove  that  cos  a  =  cot  A  =  Vcos  c. 

33.  In  a  right  spherical  triangle  prove  that 

cosM  X  sin^c  =  sin  (c  —  a)  sin  (c  +  a). 

34.  In  a  right  spherical  triangle  prove  that 

tan  a  cos  c  =  sin  h  cot  B. 

35.  In  a  right  spherical  triangle  prove  that 

sin^  A  =  cos^  B  -\-  sin^  a  sin^  B. 

36.  In  a  right  spherical  triangle  prove  that 

sin  (b-{-  c)  :=2  cos^  -J-  A  cos  b  sin  c. 

37.  In  a  right  spherical  triangle  prove  that 

sin  (c  —  b)  =  2  sin^  -J  A  cos  b  sin  c. 

38.  If,  in  a  right  spherical  triangle,  p  denotes  the  arc  of  the 
great  circle  passing  through  the  vertex  of  the  right  angle  and 
perpendicular  to  the  hypotenuse,  m  and  n  the  segments  of  the 
hypotenuse  made  by  this  arc  adjacent  to  the  legs  a  and  b, 
prove  that     (i.)  tan^  a  =  tan  c  tan  m,    (ii.)  sin^^  =  tan  m  tan  n. 


the  eight  spherical  triangle.  149 

§  52.    Solution  of  the  Isosceles  Spherical  Triangle. 

If  an  arc  of  a  great  circle  is  passed  through  the  vertex  of  an 
isosceles  spherical  triangle  and  the  middle  point  of  its  base,  the 
triangie  will  be  divided  into  two  symmetrical  right  spherical 
triangles.  In  this  way  the  solution  of  an  isosceles  spherical 
triangle  may  be  reduced  to  that  of  a  right  spherical  triangle. 

In  a  similar  manner  the  solution  of  a  regular  spherical 
polygon*  may  be  reduced  to  that  of  a  right  spherical  triangle. 
Arcs  of  great  circles,  passed  through  the  centre  of  the  polygon 
and"  its  vertices,  divide  it  into  a  series  of  equal  isosceles  tri- 
angles ;  and  each  one  of  these  -may  be  divided  into  two  equiv- 
alent right  triangles. 

Exercise  XXXIII. 

1.  In  an  isosceles  spherical  triangle,  given  the  base  h  and 
the  side  a  ;  find  A  the  angle  at  the  base,  B  the  angle  at  the 
vertex,  and  h  the  altitude. 

2.  In  an  equilateral  spherical  triangle,  given  the  side  a ; 
find  the  angle  A. 

3.  Given  the  side  a  of  a  regular  spherical  polygon  of  n 
sides ;  find  the  angle  A  of  the  polygon,  the  distance  R  from 
the  centre  of  the  polygon  to  one  of  its  vertices,  and  the  dis- 
tance r  from  the  centre  to  the  middle  point  of  one  of  its  sides. 

4.  Compute  the  dihedral  angles  made  by  the  faces  of  the 
five  regular  polyhedrons. 

5.  A  spherical  square  is  a  regular  spherical  quadrilateral. 
Find  the  angle  A  of  the  square,  having  given  the  side  a. 

*  A  regular  spherical  polygon  is  the  polygon  formed  by  the  intersec- 
tions of  the  spherical  surface  by  the  faces  of  a  regular  pyramid  whose 
vertex  is  at  the  centre  of  the  sphere. 


CHAPTER   VIII. 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


0 


§  53.    Fundamental  Formulas. 

Let  ABC  (Fig.  45)  be  an  oblique  spherical  triangle,  a,  b,  c 

its  three  sides,  A,  B,  C  the  angles 
opposite  to  them,  respectively. 
Through  C  draw  an  arc  CD 
of  a  great  circle,  perpendicular 
to  the  side  AB,  meeting  AB  at 
D.  For  brevity  let  CD  =  p, 
AD  =  m,  BD  =  7i,  ZACD  =  x, 
ZBCD  =  7/. 

1.    By  §  49  [39],  in  the  right 
triangles  BBC  and  ABC, 

sinj9  =  sin  asin  B, 
and        sin  p  =  sin  b  sin  A. 
sin  a  sin  B  =  sin  b  sin  A  ^ 
sin  a  sin  C  =  sin  c  sin  A  I  [44] 

sin b sin C  =  sine  sin B J 


Therefore, 

similarly, 

and 


These  equations  may  also  be  written  in  the  form  of 
proportions 

sin  a  :  sin  b  :  sin  c  =  sin  ^  :  sin  J?  :  sin  C. 

That  is,  the  sines  of  the  sides  of  a  spherical  triangle  are 
proportional  to  the  sines  of  the  opposite  angles. 

In  Fig.  45  the  arc  of  the  great  circle  CD  cuts  the  side  AB 
within  the  triangle.  In  case  it  cuts  AB  produced  without  the 
triangle,  sin  (180°-^),  sin  (180°— ^),  or  sin (180°- (7),  would 


THE    OBLIQUE    SPHERICAL    TRIANGLE.  151 

be  employed  in  the  above  proof  instead  of  sin^,  sin^,  or 
sin  C.  These  sines,  however,  are  equal  to  sin  A,  sin  B,  and 
sin  C,  respectively,  so  that  the  Formulas  [44]  hold  true  in  all 
cases. 

2.    In  the  right  triangle  IWC,  by  §  49  [38], 

cos  a  =  cos  j9  cos  n  =  cos^  cos  (c  —  m), 
or  (§  28)  cos  a  =  cosp  cos  c  cos  m  +  cos^  sin  c  sin  m. 

Now,  [38]  cos^  cos  m  =  cosbj 
whence  cos^  ^  cos  &  seem, 

and  cos^sinm  =  cosft  tanm 

[40]  =  cos  b  tan  b  cos  A 

=  sin  b  cos  A. 

Substituting  these  values  of  cos^cosm  and  cos^sinm  in 
the  value  of  cos  a,  we  obtain 

cos  a  =  COS  b  cos  c  +  sin  b  sin  c  cos  A"j 
and  similarly,     cos  b  =  cos  a  cos  c  +  sin  a  sin  c  cos  B  >        [45] 
cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C  J 


3.    In  the  right  triangle  ADC,  by  [41], 

gos  A  =  cosp  sin  x  =  cosp  sin  (C  —  y), 
or  (§  28)  cos  A  =  cosp  sin  (7 cos  y  —  cosp  cos  Csin  y. 

Now,  [41]  cosp  sin  y  =  cos  B ; 
whence,  cosp  ^  cos^cscy, 

and  cos p  cos  y  =  cos  B  cot  y 

[43]  =  cos  B  tan  B  cos  a 

=  sin  B  cos  a. 

Substituting  these  values  of  cosp  sin  y  and  cosp  cos  y  in  the 
value  of  cos  A,  we  obtain 

cos  A  =  —  cos  B  cos  C  +  sin  B  sin  C  cos  a~^ 
and  similarly,    cos  B  —  —  cos  A  cos  C  +  sin  A  sin  C  cos  b  V  [46] 
cosC  =  —  cos  A  cos  B  +  sin  A  sin  B  cose  J 


152 


SPHERICAL    TRIGONOMETRY. 


Formulas  [45]  and  [46]  are  also  universally  true  ;  for  the 
same  equations  are  obtained  when  the  arc  CD  cuts  the  side 
AB  without  the  triangle. 

Exercise  XXXIV. 

1.  What  do  Formulas  [44]  become  if  ^  =  90°  ?  if  ^  =  90°  ? 
if  C=90°?  if  a  =  90°?  if  ^  =  ^  =  90°?  if  a  ==^  =  90°  ? 

2.  What  does  the  first  of  [45]  become  if  ^===0°?  if  ^=90°? 
if  Jt  =  180°? 

3.  From  Formulas  [45]  deduce  Formulas  [46],  by  means  of 
the  relations  between  polar  triangles  (§  48). 

§  54.    Formulas  for  the  Half  Angles  and  Sides. 
From  the  first  equation  of  [45], 

cos^ 


I? 
cos  a  —  cos  h  cos  c 


whence, 


1  —  cos  ^  = 


sin  b  sin  c        ' 
sin  h  sin  c  +  cos  h  cos  c  —  cos  a 


sin  b  sin  c 

cos  (b  —  c)  —  cos  a  ^ 

sin  b  sin  c         ' 

.    ,  .       sin  b  sin  c  —  cos  b  cos  c  4-  cos  a 

1  +  cos  -4  — : — --; ' 

sm  b  sm  c 

cos  a  -^  cos  (b  +  c) 

sin  b  sin  c 

Hence,  by  §  30  [16]  and  [17],  and  §  31  [23], 

sin^l"  A  ==  sin  i(a-\-b  —  c)  sin  ^(a  —  b-\-c)  esc  b  esc  c, 
cos^^  A  =  sin  ^(a-\-b-\-c)  sin  ^(b-\-c—a)  esc  b  esc  c. 

Now  let  ^(a-{-b-^c)  =  s; 

whence,  i(b-\-c  —  a)  =  s  —  a, 

i(a—b-\-c)  =  s  —  b, 
^(a-^b  —  c)  =  s  —  c. 


THE    OBLIQUE    SPHERICAL    TRIANGLE.  153 

Then,  by  substitution  and  extraction  of  the  square  root, 


sin  ^  A  =  Vsin  (s  —  b)  sin  (s  —  c)  esc  b  esc  c 
cos  ^  A  =  Vsin  s  sin  (s  —  a)  esc  b  esc  c 


tan|  A  =  Vese  s  esc  (s  —  a)  sin  (s  —  b)  sin  (s  —  e) 
In  like  manner,  it  may  be  shown  that 


sin  1^  B  =  Vsin  (s  —  a)  sin  (s  —  c)  esc  a  esc  e 

cos  ^  B  =  Vsin  s  sin  (s  —  b)  esc  a  esc  e 

tan  -j-  B  =  Vese  s  esc  (s — b)  sin  (s  —  a)  sin  (s  —  e) 

sin  1^  C  =  Vsin  (s  —  a)  sin  (s  —  b)  esc  a  esc  b 

cos  J  C  =  Vsin  s  sin  (s  —  e)  esc  a  esc  b 

tan  i  C  =  Vese  s  esc  (s  —  c)  sin  (s  —  a)  sin  (s  —  b) 

Again,  from  the  first  equation  of  [46], 


whence, 


cos  a 


C0S(X  = 


1  -}-  cos  <x  = 


COS  B  cos  C  +  COS  ^ 
sin^  sin  C         ' 

sin  B  sin  C  —  cos  B  cos  C  —  cos  A 
sin^sinC 

sin  B  sin  C  +  cos  B  cos  C  +  cos  A 


sm  B  sm  C 


[47] 


If  we  place  \{A-\-B -{■  C)^S,  and  proceed  in  the  same 
manner  as  before,  we  obtain  the  following  results : 


sin  ^  a  =  V— cos  S  cos  (S  —  A)  esc  B  esc  C 


cos  i  a  =  Veos  (S  —  B)  cos  (S  —  C)  esc  B  esc  C 


tania==  V—  cos  S  cos  (S  — A)  see  (S  —  B)  sec  (S  —  C) 


[48] 


154 


SPHERICAL    TRIGONOMETRY. 


And,  in  like  manner, 


sin  I  b  =  V—  cos  S  cos  (S  —  B)  esc  A  esc  C 
cos  i  b  =  Vcos  (S  —  A)  cos  (S  —  C)  esc  A  esc  C 


tanj  b  =  V—  cos  S  cos  (S  —  B)  sec  (S  —  A)  sec  (S  —  C) 

sin  1^  c  =  V—  cos  S  cos  (S  —  C)  esc  A  esc  B 
cos  ^  e  =  Vcos  (S  —  A)  cos  (S  —  B)  esc  A  esc  B 
tan^ c  =  V—  cos  S  cos  (S  —  C)  sec  (S  —  A)  sec  (S  —  B) 

§  55.    Gauss's  Equations  and  Napier's  Analogies. 

By  §27  [5], 

cos  ^  (^  +  ^)  =  cos  -J-  ^  cos  ^  5  —  sin  ^AsiniB-, 

or,  by  substituting  for  cos  i  J,  cos  ^  B,  sin  ^  A,  sin  i  B,  their 
values  given  in  §  54,  and  reducing, 


cosi(A-\-B) 


Vsin  s  sin  (s  —  a) 
sin  b  sin  c 


-V 


sin  (s  —  h)  sin  (s  —  c) 


sin  h  sin  c 
sin  s  —  sin  (s  —  c) 
sin  G 


X 


X 


X 


/sin  s  sin  (s  —  b) 

'         sin  /7  sin  n. 


sm  a  sm  c 


/sin  (s — a)  sin  (5 — c) 
^  sin  a  sin  c 

/sin  (g — g)  sin  (s — b) 
^  sin  a  sin  ^ 


This  value,  by  applying  §§  29  [12],  31  [21],  and  observing 
that  the  expression  under  the  radical  is  equal  to  sin^  C,  becomes 

1  .  .   ,    --,-      2sin4-ccos('s  —  4-c)   .    ,  ^ 
cosj-(^4-^)=  y    — L_^_isin|(7; 

^  2sm^ccosJc  ^     ' 

and  this,  by  cancelling  common  factors,  clearing  of  fractions, 
and  observing  that  s  —  ^c  =  ^{a-{-b),  reduces  to  the  form 
cos  ^{A-\-B)go^^c  =  cos  i  {a  +  b)  sin  ^  C. 
By  proceeding  in  like  manner  with  the  values  of 

sini(^  +  5),     Go^i{A  —  B),     and  sini(^  — 5), 
three  analogous  equations  are  obtained. 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


155 


The  four  equations, 


cosi  (A  +  B)  cos  i  c  =  cos  |(a  +  b)  sin -J- C ' 
sin^(AH-B)  cosic  =  cos^(a— b)cos|^C 
cos^(A  — B)  sin|c  =  sin|(a  +  b)siii^C 
siii^(A  — B)  sin^c=siii|^(a  — b)  cos^C 


49] 


are  called  Gauss's  Equations. 

By  dividing  the  second  of  Gauss's  Equations  by  the  first, 
the  fourth  by  the  third,  the  third  by  the  first,  and  the  fourth 
by  the  second,  we  obtain 


2f  /rcv         tani(A  +  B): 


^^^f^^^cot^C 


cos  ^  (a  -f  b) 


taiii-(A  — B)  = 


sin  j-  (a  —  b) 
sin  ^  (a  -j-  b) 


cotiC 


'V> 


1  /     1  1.x       COS  4- (A— B)  ^ 
tani(a  +  b)^^^^|;^^^jtani 


a 


1^ 


tan  "I-  (a  —  b) 


sin^(A— B) 
sini(A+B) 


tan^^c 


f\|lM' 


[50] 


There  will  be  other  forms  in  each  case,  according  as  other 
elements  of  the  triangle  are  used. 

These  equations  are  called  Napier's  Analogies. 

In  the  first  equation  the  f actors, cos  J  (ct  —  b)  and  cot|-C  are 
always  positive  :  therefore,  tan  ^(A  -\-  B)  and  cos  i(a-\-b) 
must  always  have  like  signs.  Hence,  if  (X  -}-  J  <  180°,  and 
therefore  cos  ^(a  -\-b)>0,  then,  also,  tan  ^(A-\-  B)>  0,  and 
therefore  A  +  B  <  180°.  Similarly,  it  follows  that  if 
a  +  b>  180°,  then,  also,  A'{-B>  180°.  If  a-}-b  =  180°, 
and  therefore  cos  ^  (a-\-b)=^0,  then  tan  ^  (A  -{-  B)  =^  cg -^ 
whence  i(A-{-B)  =  90°,  and  ^  +  ^  ==  180°. 

Conversely,  it  may  be  shown  from  the  third  equation,  that 
a-\-b  is  less  than,  greater  than,  or  equal  to  180°,  according  as 
^  +  ^  is  less  than,  greater  than,  or  equal  to  180°. 


156 


SPHERICAL    TRIGONOMETRY. 


§  56.    Case  I. 

Given  two  sides,  a  and  b,  and  the  included  angle  C. 
The  angles  A  and  B  may  be  found  by  the  first  two  of 
Napier's  Analogies  ;  viz. : 

cos  ^(a  —  b) 


tan  J-  (A  +  B) 
tSini(A  —  B) 


cos  ^(a-\-  b) 
sin  ^(a  —  b) 


cot  I- a 

cot^a 


sin  ^  (a  -\-  b) 

After  A  and  B  have  been  found,  the  side  c  may  be  found 
by  [44]  or  by  [50] ;  but  it  is  better  to  use  for  this  purpose 
Gauss's  Equations,  because  they  involve  functions  of  the 
same  angles  that  occur  in  working  Napier's  Analogies.  Any 
one  of  the  equations  may  be  used;  for  example,  from  the 
first  we  have 


Q,o^\(a-\-b)    .     ,  ^ 
GOS^{A-{-B)        ^ 


Example. 


a  =73°  58' 64' 
6=  38°  45'   0' 
C=46°33'4r 
log  cos  i  (a  —  6)     =9. 97914 
log  sec  i  (a  +  6)     =  0. 25658 

log  cot  jC =  0.36626 

log  tan  i(^  +  B)=  0.60198 
log  sec  ^{A  +  B)  =  0.61515 
logcosi(a  +  6)  =9.74342 
log  sin  ^C =  9.59686 


log  cos  ^  c 


=  9.95543 
=  25°  31' 


therefore,  ^  (a  -  6)  =  17°  36' 57" 
i  (a +6)  =  56°  21' 57" 
iC  =23°  16' 50. 5' 

log  sin  |(a  —  6)     =9. 48092 
log  CSC  1  (a +6)     =0.07956 


coti  C 


=  0.36626 


log  tan  |(^  —  B)  =  9.92674 

^{A  +  B)=  Ib^bTAO.r 
\{A-B)=  40°  11' 25.6' 
A  =  116°  9'  6.3' 
B=  35°  46' 15.1' 
c=    51°   2' 


If  the  side  c  only  is  desired,  it  may  be  found  from  [45], 
without  previously  computing  A  and  B.  But  the  Formulas 
[45]  are  not  adapted  to  logarithmic  work.  Instead  of  changing 
them  to  forms  suitable  for  logarithms,  we  may  use  the  following 
method,  which  leads  to  the  same  results,  and  has  the  advantage 
that,  in  applying  it,  nothing  has  to  be  remembered  except 
Napier's  Eules : 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


157 


Make  the  triangle  (Fig.  46),  as  in  §  53,  equal  to  the  sum 
(or  the  difference)  of  two  right 
triangles.  For  this  purpose, 
through  B  (or  A,  but  not  C) 
draw  an  arc  of  a  great,  circle 
perpendicular  to  AC,  cutting 
AC  at  D.  Let  BD^p,  CD=m, 
AD  =  n;  and  mark  with  crosses 
the  given  parts. 
By  Rule  I., 

cos  C  =  tan  m  cot  a, 
whence  tan  m  =  tan  a  cos  C. 
By  Rule  II., 

cos  a  =  cos  m  cos^^,  whence  cosp=^  cos  a  sec  m. 
cos  c  =  cos  71  cos^,    whence  cos^  =  cos  c  sec  n. 
Therefore,  cos  c  sec  n  =  cos  a  sec  m ; 
or,  since       n  =  b  —  m,  cos  c  =  cos  a  sec  m  cos  (^  —  m). 

It  is  evident  that  c  may  be  computed,  with  the  aid  of 
logarithms,  from  the  two  equations 
tan  m  =  tan  a  cos  C, 
cose  =  cos  a  sec  m  cos  (&  —  m). 

Example.  Given  a  =- 97° 30 ' 20 ",&  =  55°  12 '10",  C=39°58'; 
find  c. 


Fig.  46. 


log  tan  6t  =0.88025  (?i) 
log  cos  (7=9.88447 
log  tan  771  =0.76472(71) 
m=      99°  45' 14" 
5-m  =  -44°33'    4" 


logcos  6^  =  9.11602(71) 
logcos  (b  —  m)  =  9. S52S6 

log  sec  7^^  =  0.77103(71) 
log  cose  =  9.73991 

c  =  56°  40' 20" 


Exercise  XXXY. 

1.  Write  formulas  for  finding,  by  ISTapier's  Rules,  the  side 
a,  when  b,  c,  and  A  are  given,  and  for  finding  the  side  b  when 
a,  c,  and  B  are  given. 


158 


SPHERICAL    TRIGONOMETRY. 


2.  Given  a  =  88°  12' 20",   ^>  =  124°  7' 17",    C=:50°2'l"; 
find  A  =  63°  15'  11",  B  =  132°  17'  59",  c  =  59°  4'  18". 

3.  Given  a  =  120°  55' 35",  &  =  88°  12' 20",   C  =  47°42'l"; 
find  ^  =  129°  58' 3",  ^  =  63°  15' 9",  c  =  55°  52' 40". 

4.  Given   ^»  =  63°  15' 12",    c  =  47°  42' 1",    ^  =:  59°  4' 25"; 
find  ^  =  88°  12' 24",   C=:  55°  52' 42",  a  =  50°l'40". 

5.  Given  ^»  =  69°  25' 11",  c  =  109°  46' 19",  ^  =  54°  54' 42"; 
find  jg=:  56°  11' 57",   C=rl23°21'12",  a  =  67°  13'. 

§  57.    Case  II. 

Criven  the  side  c  and  the  two  adjacent  angles  A  and  B. 
The  sides  a  and  h  may  be  found  by  the  third  and  fourth  of 
Napier's  Analogies, 


tan  ^{a-\-h) 


_Gosi(A  —  B) 
~Qosi(A-\-B) 


tan  4- (a  —  Z»)  = -r— ftr  ■    ..x 


tan  -J  c, 
tan  ^  c, 


and  then  the  angle  C  may  be  found  by  [44],  by  Napier's 
second  Analogy,  or  by  one  of  Gauss's  equations,  as,  for  instance, 
the  second,  which  gives 


^            cos  1  {a 

-I)   ou.^.. 

Example.      ^  =  107°  47'  V 

.•.^(^-J5)  =  34°24'20" 

B=    38°  58' 27''. 

n^  +  J5)  =  73°22'47" 

c=    51°  41' 14" 

^c  =  25°  50' 37" 

log  cos  |(^  -  5)  =  9.91648 

logsin^^- 5)  =  9.75208 

log  sec  H^  +  ^)  =  0.54359 

logcsc  1(^  +  5)  =  0.01854 

logtanic              =9.68517 

logtanic             =9.68517 

logtanHa+ft)     =0.14524 

log  tan  }  {a  —  h)    =  9. 45579 

logsini  (-^  +  -8)  =  9.98146 

^(a+&)  =  54°24'24.4" 

log  sec  i  (a -6)    =0.01703 

i(a-&)  =  15°56'25.6" 

log  cos  ic              =9.95423 

fa  =70°  20' 50" 

log  cos  |C             =9.95272 

]6  =38°  27' 59" 

i  C  =  26°  14'  52.5" 

C  C  =  52°  29' 45" 

THE    OBLIQUE    SPHERICAL    TRIANGLE. 


159 


If  the  angle  C  alone  is  wanted,  tlie  best  way  is  to  decompose 
the  triangle  into  two  right  triangles,  and  then  apply  Napier's 
Eules,  as  in  Case  I.,  when  the  side  c  alone  is  desired. 

Let   (Fig.   47)    ZABI)  =  x,  Z.CBD  =  y,  BD=p)   then, 

Eule  I., 

whence 
Eule  II. 

whence 

whence 
Hence 


cos  G  =  cot  X  cot  Aj 
cot  X  =  tan  A  cos  c. 


cos  ^=:  cos  ^  sin  0!!, 

cos  2^  =  cos  A  CSC  X. 

cos  C  =  COS  p  sin  y, 
cos  p  =  cos  C  CSC  y. 


cos  C  =  cos  ^  CSC  cc  sin  2/ 

=  cos  ^  CSC  £c  sin  (5  —  x). 
It  is  clear  that  C  may  be  computed  from  the  equations 
cot  X  =  tan  A  cos  c, 
cos  C  =  cos  ^  CSC  X  sin  (B  —  x). 


Example.    Given  A  =  35°  46'  15",  B  =  115°  9'  7",  c  =  51°  2 
find  a 

log  tan  ^  =  9.85760 
log  cose  =  9.79856 


log  cot  X  =9.65616 

X  =  65°  37'  35" 
.•,B-x  =49°  31' 32" 


log  cos  ^  =9.90992 

logsin  (5  — £c)  =  9.88122 
log  CSC  X  =  0.04055 
logcos  (7=  9.83099 

C=47°20'30" 


Exercise  XXXVI. 

1.  What  are  the  formulas  for  computing  A  when  B,  C,  and 
a  are  given ;  ^.nd  for  computing  B  when  A,  C,  and  b  are  given  ? 

2.  Given  A  =  26°  58'  46",  B  =  39°  45'  10",  e  =  154°  46'  48"; 
find  a  =  37°  14'  10",  h  =  121°  28'  10",  C=  161°  22'  11". 

3.  Given  A  =  128°  41'  49",  B  =  107°  33'  20",  c  =  124°  12'  31"; 
find  a  =  125°  41'  44",  h  =  82°  47'  34",  C  =  127°  22'. 


160  SPHERICAL    TRIGONOMETRY. 

4.  Given  ^=153°  17' 6",  (7=  78°  43' 36",  a  =  86°  15' 15"; 
find  &  =  152°  43'  51",  c  =  88°  12'  21",  A  =  78°  15'  48". 

5.  Given  A  =  125°  41'  44",  C=  82°  47'  35",  b  =  52°  37'  57"; 
find  a  =  128°  41'  46",  c  =  107°  33'  20",  B  =  55°  47'  40". 

§  5S.    Case  III. 
Given  two  sides  a  and  b,  and  the  angle  A  o2yposite  to  a. 
The  angle  B  is  found  from  [44],  whence  we  have 
sin  B  =  sin  A  sin  b  esc  a. 

When  B  has  been  found,  C  and  c  may  be  found  from  the 
fourth  and  the  second  of  Napier's  Analogies,  from  which  we 
obtain 

^"^^^^-sinH^-^^^"^(^-^)- 

eotiC  =  $4^^tanH^-^). 
^  sin  ^  (a  —  ^>)         ^  ^  ^ 

The  third  and  first  of  Napier's  Analogies  may  also  be  used. 

Note  1.  Since  B  is  determined  from  its  sine,  the  problem  in  general 
has  two  solutions ;  and,  moreover,  in  case  sin  B  >►  1 ,  the  problem  is 
impossible.  By  geometric  construction  it  may  be  shown,  as  in  the 
corresponding  case  in  Plane  Trigonometry,  under  what  conditions  the 
problem  really  has  two  solutions,  one  solution,  and  no  solution.  But  in 
practical  applications  a  general  knowledge  of  the  shape  of  the  triangle  is 
known  beforehand ;  so  that  it  is  easy  to  see,  without  special  investigation, 
which  solution  (if  any)  corresponds  to  the  circumstances  of  the  question. 

It  can  be  shown  that  there  are  two  solutions,  when  A  and  a  are  alike 
in  kind  and  sin  6  >  sin  a  >  sin  ^  sin  6  ;  no  solution  when  A  and  a  are 
unlike  in  kind  (including  the  case  in  which  either  ^  or  a  is  90°)  and 
sin  h  is  greater  than  or  equal  to  sin  a,  or  when  sin  a<C.sinA  sin  b ;  and 
one  solution  in  every  other  case. 

Note  2.  The  side  c  or  the  angle  C  may  be  computed,  without  first 
finding  B,  by  means  of  the  formulas 

tan  m  =  cos  A  tan  6,  and  cos  (c  —  m)  =  cos  a  sec  b  cos  m, 
cot  X  ==  tan  A  cos  &,  and  cos  (C  —  x)  =  cot  a  tan  6  cos  x. 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


161 


These  formulas  may  be  obtained  by  resolution  of  the  triangle  into  right 
triangles,  and  applying  Napier's  Rules ;  m  is  equal  to  that  part  of  the 
side  c  included  between  the  vertex  A  and  the  foot  of  the  perpendicular 
from  C,  and  x  is  equal  to  the  corresponding  portion  of  the  angle  C. 

Note  3.  After  the  two  values  of  B  have  been  obtained,  the  number 
of  solutions  may  readily  be  determined  by  §  48  —  I.  If  log  sin  B  is  posi- 
tive, there  will  be  no  solution. 

Example.     Given  a  =  57°  36',  b  =  31°  12',  A  =  104°  25'  30". 


In  this  case  A  >   90°, 

and  a+6<180°; 

therefore,  ^  +  ^<180°; 

hence,  -B<  90°, 

and  only  one  solution. 

a  +  b  ==88°  50' 
a-b  =26°26' 
A  +  -B=140°51'53" 
A-B=    67° 59'   7" 
logsini(^  +  J5)  =  9.97416 
logcsci(^-^)=  0.25252 
log  tan  Ija  —  b)    -  9. 37080 
logtanic  =  9.59748 

ic  =  21°35'38" 
c  =  43°  11' 16" 


logsinJ.  =  9.98609 
log  sin  6  =9.71435 
log  CSC  g  =0.07349 
log  sin  I?  =  9.77393 

i?  =  36°27'20" 

i(a  +  6)    =44°  25' 
^{a-b)   =  13°  13' 
^(^  +  5)  =  70°  26' 25' 
i  (^  -  ^)  =  33°  59'   5' 
log  sin  1  (a  +  b)   =  9.84502 


log  CSC  I  {a  - 
log  tan  I  (A 


b)    =0.64086 
-  B)  =  9.82873 


log  cot  10=  0.31461 

1C  =  25°51'15' 
C=  51°  42' 30' 


Exercise  XXXYII. 

1.    Given  a  =  73°  49'  38",  b  =  120°  53'  35",  A  =  88°  52'  42"; 
find  B  =  116°  42'  30",  c  =  120°  57'  27",  C  =  116°  47'  4". 

2. .  Given  a  =  150°  57'  5",  b  =  134°  15'  54",  A  =  144°  22'  42"; 
find  ^1  =  120°  47' 45",    Ci  =  55°42'8",         Ci  =  97°  42' 55.4"; 
B^=    59°  12' 15",    C2  =  23°57'17.4",    C^  =  29°    8' 39". 

3.  Given  a  =  79°  0' 54.5",  J  =  82°  17' 4",  ^  =  82°  9' 25.8"; 
find  B  =  90°;  c  =  45°  12'  19",  C  =  45°  44'. 

4.  Given  a  =  30°  52' 36.6",  J  =  31°  9' 16",  ^  =  87°  34' 12"; 
show  that  the  triangle  is  impossible. 


162  SPHERICAL    TRIGONOMETRY. 

§  59.    Case  IV. 

Given  two  angles  A  and  By  and  the  side  a  opposite  to  one  of 
them. 

The  side  b  is  found  from  [44],  whence 

sin  b  =  sin  a  sin  B  esc  A. 

The  values  of  c  and  C  may  then  be  found  by  means  of 
Napier's  Analogies,  the  fourth  and  second  of  which  give 

^      .         sm^(A  +  B)^      ,, 

sm  ^(a  —  b)         ^  ^  ^ 

Note  1.  In  this  case  the  conditions  for  one,  two,  or  no  solutions  can 
be  deduced  directly  by  the  theory  of  polar  triangles  from  the  correspond- 
ing conditions  of  Case  III.  There  are  two  solutions,  when  A  and  a  are 
alike  in  kind  and  sin  J5>  sin  ^  >  sin  a  sin  B  ;  no  solution  when  A  and  a 
are  unlike  in  kind  (including  the  case  in  which  either  tI  or  a  is  90°)  and 
sin  B  is  greater  than  or  equal  to  sin  A,  or  when  sin  J.  <  sin  a  sin  B  ;  and 
one  solution  in  every  other  case. 

Note  2.  By  proceeding  as  indicated  in  Case  III.,  Note  2,  formulas 
for  computing  c  or  C,  independent  of  the  side  b,  may  be  found  ;  viz. : 

tan  m  =  tan  a  cos  B,  and  sin  (c  —  m)  =  cot  A  tan  B  sin  m, 
cot  X  =  cos  a  tan  B,  and  sin  (C  —  ic)  =  cos  A  sec  B  sin  x. 

In  these  formulas  m  =  BD,  x—  L  BCD,  D  being  the  foot  of  the  per- 
pendicular from  the  vertex  C. 

Note  3.  As  in  Case  III. ,  only  those  values  of  b  can  be  retained  which 
are  greater  or  less  than  a,  according  as  B  is  greater  or  less  than  A.  If 
log  sin  b  is  positive,  the  triangle  is  impossible. 

Exercise  XXXVIII. 

1.  Given  A  =  110°  10',  B  =  133°  18',  a  =  147°  5'  32" ;  find 
b  =  155°  5'  18",  c  ==  33°  1'  36",  C  =  70°  20'  40". 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


163 


2.  Given  A  =  113°  39'  21",  B  =  123°  40'  18",  a  =  65°  39'  46"; 
find  b  =  124°  7'  20",  c  =  159°  50'  14",  C  =  159°  43'  34". 

3.  Given  A  =  100°  2'  11.3",  5= 98°  30'  28",  a =95°  20'  38.7"; 
find  b  =  90°,  c  =  147°  41'  43",  C  =  148°  5'  33". 

4.  Given  A  =  24°  33'  9",  B  =  38°  0'  12",  a  =  65°  20'  13" ; 
show  that  the  triangle  is  impossible. 

§  60.    Case  V. 
Given  the  three  sides,  a,  b,  and  c. 

The  angles  are  computed  by  means  of  Formulas  [47],  and 
the  corresponding  formulas  for  the  angles  ^  and  C. 

The  formulas  for  the  tangent  are  in  general  to  be  preferred. 
If  we  multiply  the  equation 


tan  ^-4  =  Vcsc  s  esc  (s  —  a)  sin  (s  —  b)  sin  (5  —  c) 

by  the  equation  1  =  -. — \ ^>   and  put 

^  ^  sin(s  — a)  ^ 


Vcsc  s  sin  (s  —  a)  sin  (s  —  b)  sin  {s  —  c)  =  tan  r, 

and  also  make  analogous  changes  in  the  equations  for  tan-J^ 
and  tan  ^  C,  we  obtain 

tan  ^A=  tan  r  esc  (s  —  a), 

tan  -J-  j5  =  tan  r  esc  (s  —  b), 

tan  i  C  =  tan  r  esc  (5  —  c), 
which  are  the  most  convenient  formulas  to  employ  when  all 
three  angles  have  to  be  computed. 


Example  1. 


a=  50°  54' 32' 

b=  37°  47' 18' 

c=  74°  51' 50' 

2 8=  163°  33'  40' 

s=  81°  46' 50' 

s-a=  30°  52' 18' 

s-b=  43°  59' 32' 

s-c=  6°  55'    0' 


log  CSC  5=    0.00448 

log  CSC  (s  — a)  =   0.28978 

log  sin  (s- 6)=    9.84171 

log  sin  (s  —  c)  =    9.08072 

2)19.21669 

log  tan  i^  =   9.60835 

1^  =  22°    5' 20" 
A  =  44°  10'  40" 


164 


SPHERICAL    TRIGONOMETRY. 


Example  2.     a  -  124°  12'  31" 

s 

-a=    13°  39'    5" 

b=    54°  18' 16" 

s 

-b=    83°  33'  20" 

c=    97°  12' 25" 

s 

-  c=    40°  39'  11" 

2  s  =  275°  43'  12" 

log 

tan 

^A  =  0.30577 

s  =  137°  51'  36" 

log 

tan 

15=9.68145 

log  sin  {s  —  a)  =  9.37293 

log 

tan 

hC=  9.86480 

logsin(s- 6)  =9.99725 

IA=    63°  41'    3.8" 

logsin(s-  c)  =  9.81390 

^B=    25°  39'    5.6" 

log  CSC  5=0.17331 

iC=    36°  13' 20.1" 

log  tan2r=  9.35739 

A  =  127°  22'    7" 

logtanr  =  9.67870 

B=    61°  18' 11" 
C=    72°  26' 40" 

Exercise  XXXIX. 

1.  Given  a  =  120°  55' 35",  ^>  =  59°4'25",  c  =  106°  10' 22"; 
find  A  =  116°  44'  50",  B  =  63°  15'  18",  C  =  91°  7'  22". 

2.  Given  a  =  50°  12' 4",  ^»  =  116°  44' 48",  c  =  129°  11' 42"; 
find  A  =  59°  4'  28",  B  =  94°  23'  12",  C  =  120°  4'  52". 

3.  Given  a  =  131°  35' 4",  ^»  =  108°  30' 14",  c  =  84°  46' 34"; 
find  ^  =  132°  14' 21",  ^  =  110°  10' 40",  (7  =  99°  42' 24". 

4.  Given  a  =  20°  16' 38",   ^'  =  56°  19' 40",   c  =  66°  20' 44"; 
find  A  =  20°  9'  54",  B  =  55°  52'  31",  C  =  114°  20'  17". 


§  61.    Case  VI. 
Given  the  three  angles,  A,  B,  and  C. 

The  sides  are  computed  by  means  of  Formulas  [48].     The 
formulas  for  the  tangents  are  in  general  to  be  preferred. 
If  we  multiply  the  equation 

tan  |-<z  =  \l—  cos  *S^  cos  {S—A)  sec  {S  —  B)  sec  {S  —  C) 


by  the  equation  1  = f^ -J- 


seG(S  —  Ay 


and  put 


V—  COS  S sec (S  —  A)  sec  (S  —  B)  sec  {S—C)=  tan R, 
and  also  make  analogous  changes  in  the  equations  for  tan|-6 
and  tan  ^c,  we  obtain 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


165 


tan  "l-a  =  tan  i^  COS  (aS^  —  ^), 

tan  ^  &  =  tan  ^  cos  (iS' —  ^), 

tan  ic=  tan  B  cos  (S  —  C), 
which  are  the  most  convenient  formulas  to  use  in  case  all 
three  sides  have  to  be  computed. 

In  Example  1,  after  we  find  the  values  of  S,  S  —  A,  S—  B, 
S — C,  we  write  the  formula  for  tan^a  with  the  algebraic 
sign  written  above  each  function  as  follows : 


tan  i a  =  -W—  cos  *S^  cos  (S  —  A)  sec  (S 

Example  1.  a  =  220° 
B  =  180° 
C  =  150° 


B)seG(S-C). 


2S=  500° 

S  =  250° 

S-A=    30° 

S-  B  =  120° 

S-  C  =  100° 


logcosS  =  9.53405  (n) 
logcos(S-^)  =  9.93753 
log  sec  {S-  B)  =  0.30103  (n) 
log  sec  {S  -  C)  =  0.76033  (n) 
2)0.53294 
logtania  =  0.26647 

ia=    61° 34'  6'' 
a  =  123°   8' 12" 


Note.  Here  the  effect,  as  regards  algebraic  sign,  of  three  negative 
factors,  is  cancelled  by  the  negative  sign  belonging  to  the  whole  product. 

In  Example  2,  after  we  find  the  values  of  S,  S  —  A,  S~B, 
S—  C,we  write  the  formula  for  tan  B  with  the  algebraic  sign 
written  above  each  function  as  follows  : 


tan^=: 
Example  2. 


nI- 


cos  S  sec  (S  -  A)  sec  (S  -  B)  sec  {S-C). 


A=    20°   9' 56'' 
B=    55°  52' 32" 
C=  114°  20' 14" 
2S=  190°  22'  42" 
logcosS  =  8.95638  (n) 
logsec(/S- -4)  =  0.58768 
logsec(S-JB)  =  0.11143 
log  sec  (S-C)  =  0.02472 
log  tan2E  =  9.68021 
log  tan  E  =9.84010 


S  = 

95°  11' 21" 

s- 

-A  = 

75°    1'25' 

s- 

-  B  = 

39°  18'  49' 

s- 

-C  = 

-19°    8' 53' 

logtan 

la  = 

9.25242 

log  tan 

hb  = 

9.72867 

log  tan 

\o  = 

9.81538 

ia  = 

10°   8' 18.9" 

\^  = 

28°   9' 50.4" 

^.c  = 

33°  10' 21.3" 

a  = 

20°  16'  38" 

b  = 

56°  19'  41" 

c  = 

66°  20'  43" 

166  SPHEKICAL    TRIGONOMETRY. 

Exercise  XL. 

1.  Given  A  =  130°,  B  =  110°,  (7  =  80° ; 

find  a  =  139°  21 '  22",  b  =  126°  57'  52",  c  =  56°  51'  48". 

2.  Given  J^  =  59°  55'  10",  B  =  85°  36'  50",  C  =  59°  55'  10"; 
find  a  =  51°  17'  31",  6  =  64°  2'  47",  c  ^  51°  17'  31". 

3.  Given  A  =  102°  14'  12",  B  =  54°  32'  24",  C  =  89°  5'  46"; 
find  a  =  104°  25' 9",  ^>  =  53°  49' 25",  c  =  97°  44' 19". 

4.  Given  J  =  4°  23' 35",  ^  =  8°  28' 20",  C=  172°  17' 56"; 
find  a  =  31°  9'  14",  b  =  84°  18'  2S",  c  =  115°  10'. 

§  62.    Area  of  a  Spherical  Triangle. 

I.    When  the  three  angles,  A,  B,  C,  are  given. 
Let        R  =  radius  of  sphere, 

E  =  the  spherical  excess  =  A-\-B-\-  C— 180°, 

i^=area  of  triangle. 

Three  planes  passed  through  the  centre  of  a  sphere,  each 
perpendicular  to  the  other  two  planes,  divide  the  surface  of 
the  sphere  into  eight  tri-rectangular  triangles. 

It  is  convenient  to  divide  each  of  these  eight  triangles  into 
90  equal  parts,  and  to  call  these  parts  spherical  degrees.  The 
surface  of  every  sphere,  therefore,  contains  720  spherical 
degrees. 

Since  in  spherical  degrees,  the  AABC=E,  and  the  entire 
surface  of  the  sphere  is  equal  to  720  spherical  degrees, 

.-.  A  ABC:  surface  of  the  sphere  =  ^  :  720; 
or,  since  the  surface  of  a  sphere  =  AttB^, 

A  ABC:  4.'7rR''  =  E:  720; 
whence 


THE    OBLIQUE    SPHERICAL    TRIANGLE.  167 

II,    When  the  thi^ee  sides,  a,  b,  c,  are  given. 

A  formula  for  computing  the  area  is  deduced  as  follows : 

From  the  first  of  [49], 

cos  ^{A-\-  B)  _  cos  -J-  (<x  +  ^)  . 

cos(90°— iC)~       cos-Jc       ' 

whence,  by  the  Theory  of  Proportions, 

cosj  {A-\-B)  —  cos  (90°— i  C)  _  cosi(a+^)-cos^c 
cosi(^+^)  +  cos(90°-^C)~"cosi(a+^»)+cos^c'    ^^^ 

Now,  in  §  31,  the  division  of  [23]  by  [22]  gives 

cos  ^  —  cos  5  ,       ,  /  .    ,    T>s  ,       ,  /  .       T..  /,  s 

—^^—^  =  -t^u^^iA  +  B)Ur.U^-B),  (b) 

in  which  for  A  and  B  we  may  substitute  any  other  two 
angular  magnitudes,  as  for  example,  ^(A-\-B)  and  (90°— -J-  C), 
or  ^((i-\-h)  and  J  c. 

If  we  use  in  place  of  A  and  B  the  values  ^{A-\-B)  and 
(90°  —  ^  C),  the  first  side  of  equation  (b)  becomes 
cos  ^{A-\-B)  —cos  (90°  —  ^C). 
cosi(^  +  ^)  +  cos(90°-i(7)' 

and  the  second  side  becomes 

-tani(|^+i^+90°-iC)tani(i^+i^-90°  +  |C); 
or, 

-tani(^  +  ^- (7+180°)  tan  J(^  +  ^+C-180°). 
If  we  remember  that  £'=^  +  ^+ C  — 180°,  and  observe 
that 

tani(^+^-C+180°)  =  tani(360°-2(7+^+^+C-180°) 
=  tani(360°-2(7+^) 
=  tan[90°-i(2C-^)] 
=  coti(2C-^), 

it  will  be  evident  that  equation  (b)  may  be  written 
cosi(^+J?)-cos(90°-iC)^_ 
oosi(^  +  5)+cos(90°-jC)  *"-  ''      *      ^> 


168  SPHERICAL    TRIGONOMETRY. 

If  we  substitute,  in  equation  (b),  for  A  and  B,  the  values 
^{a-\-h)  and  \  c,  and  also  substitute  s  ioi:  ^  {a -\- h -\- c)  and 
s  —  c  for  i-  (a  +  ^  —  c),  equation  (b)  will  become 

cos  ^  (a -f  ^)  —  cos  ^  c  ^      ,     ^  .         ^  ,,. 

^ — ^— f- f-  =  — tani5tani(s  — c).  (d) 

cos  ^  (c)^  -}-  ^)  +  cos  I  c  ^  2  ^         ^  ^  ^ 

Comparing  (a),  (c),  and  (d),  we  obtain 

cot  i  (2  C—  ^)  tan  i  ^= tan  ^  s  tan  ^  {s  —  c).  (e) 

By  beginning  with  the  second  equation  of  [49],  and  treating 
it  in  the  same  way,  we  obtain  as  the  result, 

tani  (2  C—E)  tanj  JS'^tan^  (5  —  a)  tan  i  (s  —  h).         (f) 

By  taking  the  product  of  (e)  and  (f),  we  obtain  the  elegant 
formula, 

tan2iE=tan^staii^(s— a)tan|(s— b)tan^(s— c),    [52] 
which  is  known  as  I'Huilier's  Formula. 

By  means  of  it  E  may  be  computed  from  the  three  sides, 
and  then  the  area  of  the  triangle  may  be  found  by  [51]. 

III.  In  all  other  cases,  the  area  may  be  found  by  first 
solving  the  triangle  so  far  as  to  obtain  the  angles  or  the  sides, 
whichever  may  be  more  convenient,  and  then  applying  [51] 
or  [52]. 


Example  1.   A  =  102^  14'  \2" 
B=    54°  32' 24" 

C=    89°  5' 46'' 


245°  52' 22' 
E=    65°  52' 22' 
=  237142" 
180°  =  648000" 


logE2  =  logE2. 
log^  =5.37501 


logF  =0.06058  f  logE2 
F=  1.1497^2 


If,  therefore,  we  know  the  radius  of  the  sphere,  we  can 
express  the  area  of  a  spherical  triangle  in  the  ordinary  units 
of  area. 

*  See  Wentworth  &  HUl's  Tables,  page  20. 


THE    OBLIQUE    SPHERICAL    TRIANGLE. 


169 


Example  2. 


a  =133°  26^9'' 
b=  64°  50' 53'' 
c-  144°  13' 45" 

2s  =  342°  30' 57" 
s=  171°  15' 28. 5' 

-a=   37° 49'  9.5' 

-  6=  106°  24' 35. 5' 

-  c=   27°   1'43.5' 


is=85°37'44' 
i(s-a)  =  18°54'35' 
1  (s  -  6)  =  53°  12'  18' 
1(8 -c)  =  13°  30' 52' 
log  tan  is  =  1.11669 
logtaiii(s- a)  =  9.53474 
logtani(s-  6)  =  0.12612 
log  tan  1  (g  —  c)  =  9.38083 
logtan2^^  =  0.15838 
0.07919 
50°  11' 
^  =  200°  46' 52' 


Exercise  XLI. 

1.  Given  ^  =  84°  20' 19",  ^  =  27°  22' 40",  (7  =  75°  33'; 
find  ^=26159",  F=0.126S2E^ 

2.  Given  a  =  69°  15'  6",  b  =  120°  42'  47",  c  =  159°  18'  33"; 
find  ^=216°  40' 28". 

3.  Given  a  =  33°  V  45",  b  =  155°  5'  18",  C  =  110°  10' ; 
find  ^=133°  48' 53". 

4.  Find  the  area  of  a  triangle  on  the  earth's  surface 
(regarded  as  spherical),  if  each  side  of  the  triangle  is  equal 
to  1°.     (Eadius  of  earth  =  3958  miles.) 


CHAPTER   IX. 


APPLICATIONS  OF  SPHERICAL  TRIGONOMETRY. 


§  63.    Problem. 


To  reduce  an  angle  measured  in  space  to  the  horizon. 
Let  0  (Pig.  48)  be  the  position  of  the  observer  on  the  ground, 

AOB  =  h,t\iQ  angle  measured  in 
space,  (for  example,  the  angle 
between  the  tops  of  two  church 
spires),  OA^  and  OB'  the  projec- 
tions of  the  sides  of  the  angle 
upon  the  horizontal  plane  HR, 
AOA'  =  m  and  BOB'  =  n,  the 
angles  of  inclination  of  OA  and 
OB  respectively  to  the  horizon. 
Required  the  angle  A'OB' =  x 
made  by  the  projections  on  the 
horizon. 
The  planes  of  the  angles  of  inclination  AOA'  and  BOB' 
produced  intersect  in  the  line  OC,  which  is  perpendicular  to 
the  horizontal  plane  (Geom.  §  520). 

From  0  as  a  centre  describe  a  sphere,  and  let  its  surface  cut 
the  edges  of  the  trihedral  angle  0-ABC  in  the  points  M,  iV, 
and  P.  In  the  spherical  triangle  MNF  the  three  sides 
MN=  h,  MP  =  90°  —  m,  NP  =  90°  —  n,  are  known,  and  the 
spherical  angle  P  is  equal  to  the  required  angle  x. 
From  §  48  we  obtain 

cos  ^x=  Vcos  s COS  (s  —  h)  sec  m  sec  n, 
where  ^  (ni  -\- n -{■  h)  =^  s. 


Fig.  48. 


APPLICATIONS. 


171 


§  64.   Problem. 

To  find  the  distance  between  two  places  on  the  earth^s  surface 
(regarded  as  spherical),  given  the  latitudes  of  the  places  and  the 
difference  of  their  longitudes. 

Let  M  and  iV  (Fig.  49)  be  the  places ;  then  their  distance 
MN  is  an  arc  of  the  great  circle 
passing  through  the  places.  Let 
F  be  the  pole,  AB  the  equator. 
The  arcs  MB  and  NS  are  the 
latitudes  of  the  places,  and  the 
arc  BS,  or  the  angle  MPJSf,  is 
the  difference  of  their  longi- 
tudes. Let  MB  =  bj  JSfS  =  aj 
BS  =  I ;  then  in  the  spherical 
triangle  MNF  two  sides,  MF 
=90°— b,  NF=90°  —  a,  and  the 
included  angle  MFN=  I,  are  given,  and  we  have  (from  §  56) 

tan  m     =  cot  a  cos  I, 

cos  MN^  sin  a  sec  m  sin  (b-\-rri). 

From  these  equations  first  find  m,  then  the  arc  MN,  and 
then  reduce  MN  to  geographical  miles,  of  which  there  are  60 
in  each  degree. 


Fig.  49. 


§  Q)^.    The  Celestial  Sphere. 

The  Celestial  Sphere  is  an  imaginary  sphere  of  indefinite 
radius,  upon  the  concave  surface  of  which  all  the  heavenly- 
bodies  appear  to  be  situated. 

The  Celestial  Equator,  or  Equinoctial,  is  the  great  circle  in 
which  the  plane  of  the  earth's  equator  produced  intersects  the 
surface  of  the  celestial  sphere. 

The  Poles  of  the  equinoctial  are  the  points  where  the  earth's 
axis  produced  cuts  the  surface  of  the  celestial  sphere. 


172  SPHERICAL    TRIGONOMETRY. 

The  Celestial  Meridian  of  an  observer  is  the  great  circle  in 
which  the  plane  of  his  terrestrial  meridian  produced  meets 
the  surface  of  the  celestial  sphere. 

Hour  Circles,  or  Circles  of  Declination,  are  great  circles 
passing  through  the  poles,  and  perpendicular  to  the  equinoctial. 

The  Horizon  of  an  observer  is  the  great  circle  in  which  a 
plane  tangent  to  the  earth's  surface,  at  the  place  where  he  is, 
meets  the  surface  of  the  celestial  sphere. 

The  Zenith  of  an  observer  is  that  pole  of  his  horizon  which 
is  exactly  above  his  head. 

Vertical  Circles  are  great  circles  passing  through  the  zenith 
of  an  observer,  and  perpendicular  to  his  horizon. 

The  vertical  circle  passing  through  the  east  and  west  points  of 
the  horizon  is  called  the  Prime  Vertical ;  that  passing  through 
the  north  and  south  points  coincides  with  the  celestial  meridian. 

The  Ecliptic  is  a  great  circle  of  the  celestial  sphere, 
apparently  traversed  by  the  sun  in  one  year  from  west  to  east, 
in  consequence  of  the  motion  of  the  earth  around  the  sun. 

The  Equinoxes  are  the  points  where  the  ecliptic  cuts  the 
equinoctial.  They  are  distinguished  as  the  Vernal  equinox 
and  the  Autumnal  equinox  ;  the  sun  in  his  annual  journey 
passes  through  the  former  on  March  21,  and  through  the 
latter  on  September  21. 

Circles  of  Latitude  are  great  circles  passing  through  the 
poles  of  the  ecliptic,  and  perpendicular  to  the  plane  of  the 
ecliptic. 

The  angle  which  the  ecliptic  makes  with  the  equinoctial  is 
called  the  obliquity  of  the  ecliptic ;  it  is  equal  to  23°  27', 
nearly,  and  is  often  denoted  by  the  letter  e. 

These  definitions  are  illustrated  in  Figs.  50  and  51.  In 
Fig.  50,  AVBUi^  the  equinoctial,  P  and  P'  its  poles,  NPZS 
the  celestial  meridian  of  an  observer,  NESW  his  horizon,  Z 
his  zenith,  31  a  star,  PMP'  the  hour  circle  passing  through 
the  star,  ZMDZ'  the  vertical  through  the  star. 


APPLICATIONS. 


173 


In  rig.  51,  AVBU  represents  the  equinoctial,  EVFU  the 
ecliptic,  P  and  Q  their  respective  poles,  F  the  vernal  equinox, 
?7  the  autumnal  equinox,  M  a  star,  PMB  the  hour  circle 
through  the  star,  QMT  the  circle  of  latitude  through  the  star, 
andZ^r^^e. 


Fig.  51. 


The  earth's  diurnal  motion  causes  all  the  heavenly  bodies 
to  appear  to  rotate  from  east  to  west  at  the  uniform  rate  of 
15°  per  hour.  If  in  Fig.  50  we  conceive  the  observer  placed  at 
the  centre  0,  and  his  zenith,  horizon,  and  celestial  meridian 
fixed  in  position,  and  all  the  heavenly  bodies  rotating  around 
PP^  as  an  axis  from  east  to  west  at  the  rate  of  15°  per  hour, 
we  form  a  correct  idea  of  the  apparent  diurnal  motions  of 
these  bodies.  When  the  sun  or  a  star  in  its  diurnal  motion 
crosses  the  meridian,  it  is  said  to  make  a  transit  across  the 
meridian  ;  when  it  passes  across  the  part  NWS  of  the  horizon, 
it  is  said  to  set;  and  when  it  passes  across  the  part  NES,  it  is 
said  to  rise  (the  effect  of  refraction  being  here  neglected). 
Each  star,  as  M,  describes  daily  a  small  circle  of  the  sphere 
parallel  to  the  equinoctial,  and  called  the  Diurnal  Circle  of 
the  star.  The  nearer  the  star  is  to  the  pole  the  smaller  is  the 
diurnal  circle ;  and  if  there  were  stars  at  the  poles  P  and  P\ 
they  would  have  no  diurnal  motion.     To  an  observer  north  of 


174  SPHERICAL    TRIGONOMETRY. 

the  equator,  the  north  pole  P  is  elevated  above  the  horizon 
(as  shown  in  Fig.  50)  ;  to  an  observer  south  of  the  equator, 
the  south  pole  P'  is  the  elevated  pole. 

§  QQ.    Spherical  Co-ordinates. 

Several  systems  of  fixing  the  position  of  a  star  on  the  sur- 
face of  the  celestial  sphere  at  any  instant  are  in  use.  In  each 
system  a  great  circle  and  its  pole  are  taken  as  standards  of 
reference,  and  the  position  of  the  star  is  determined  by  means 
of  two  quantities  called  its  spherical  co-ordinates. 

I.  If  the  horizon  and  the  zenith  are  chosen,  the  co-ordinates 
of  the  star  are  called  its  altitude  and  its  azimuth. 

The  Altitude  of  a  star  is  its  angular  distance,  measured  on 
a  vertical  circle,  above  the  horizon.  The  complement  of  the 
altitude  is  called  the  Zenith  Distance. 

The  Azimuth  of  a  star  is  the  angle  at  the  zenith  formed  by 
the  meridian  of  the  observer  and  the  vertical  circle  passing 
through  the  star,  and  is  measured  therefore  by  an  arc  of  the 
horizon.  It  is  usually  reckoned  from  the  north  point  of  the 
horizon  in  north  latitudes,  and  from  the  south  point  in  south 
latitudes ;  and  east  or  west  according  as  the  star  is  east  or 
west  of  the  meridian. 

II.  If  the  equinoctial  and  its  pole  are  chosen,  then  the 
position  of  the  star  may  be  fixed  by  means  of  its  declination 
and  its  hour  angle. 

The  Declination  of  a  star  is  its  angular  distance  from  the 
equinoctial,  measured  on  an  hour  circle.  The  angular  distance 
of  the  star,  measured  on  the  hour  circle,  from  the  elevated  pole, 
is  called  its  Polar  Distance. 

The  declination  of  a  star,  like  the  latitude  of  a  place  on  the 
earth's  surface,  may  be  either  north  or  south ;  but,  in  practical 
problems,  while  latitude  is  always  to  be  considered  positive, 
declination,  if  of  a  different  name  from  the  latitude,  must  be 
regarded  as  negative. 


APPLICATIONS.  175 

If  the  declination  is  negative,  the  polar  distance  is  equal 
numerically  to  90°  +  the  declination. 

The  Hour  Angle  of  a  star  is  the  angle  at  the  pole  formed 
by  the  meridian  of  the  observer  and  the  hour  circle  passing 
through  the  star.  On  account  of  the  diurnal  rotation,  it  is 
constantly  changing  at  the  rate  of  15°  per  hour.  Hour  angles 
are  reckoned  from  the  celestial  meridian,  positive  towards  the 
west,  and  negative  towards  the  east. 

III.  The  equinoctial  and  its  pole  being  still  retained,  we 
may  employ  as  the  co-ordinates  of  the  star  its  declination  and 
its  right  ascension. 

The  Eight  Ascension  of  a  star  is  the  arc  of  the  equinoctial 
included  between  the  vernal  equinox  and  the  point  where  the 
hour  circle  of  the  star  cuts  the  equinoctial.  Right  ascension  is 
reckoned  from  the  vernal  equinox  eastward  from  0°  to  360°. 

IV.  The  ecliptic  and  its  pole  may  be  taken  as  the  standards 
of  reference.  The  co-ordinates  of  the  star  are  then  called  its 
latitude  and  its  longitude. 

The  Latitude  of  a  star  is  its  angular  distance  from  the 

ecliptic  measured  on  a  circle  of  latitude. 

The  Longitude  of  a  star  is  the  arc  of  the  ecliptic  included 

between  the  vernal  equinox  and  the  point  where  the  circle  of 

latitude  through  the  star  cuts  the  ecliptic. 
For  the  star  M  (Fig.  50), 

let  '         1  =  the  latitude  of  the  observer, 

h  =  DM       =  the  altitude  of  the  star, 
z  =  ZM        =  the  zenith  distance  of  the  star, 
a  =  /_PZM=^  the  azimuth  of  the  star, 
t=^  Z.  ZPM^  the  hour  angle  of  the  star, 
d  =  EM        =  the  declination  of  the  star, 
p  =  FM        =  the  polar  distance  of  the  star, 
r  =  VR         =  the  right  ascension  of  the  star, 
u  =  MT        =the  latitude  of  the  star  (Fig.  51), 
v=VT         =  the  longitude  of  the  star  (Fig.  51). 


176 


SPHERICAL    TRIGONOMETRY. 


In  many  problems,  a  simple  way  of  representing  the  mag- 
nitudes involved,  is  to  project  the  sphere  on  the  plane  of  the 

horizon,  as  shown  in  Fig.  52. 

NESW  is  the  horizon,  Z 
the  zenith,  NZS  the  meridian, 
WZE  the  prime  vertical,  WAE 
the  equinoctial  projected  on  the 
plane  of  the  horizon,  P  the 
elevated  pole,  M  a  star,  DM 
its  altitude,  ZM  its  zenith  dis- 
tance, Z  FZM  its  azimuth,  MR 
its  declination,  PM  its  polar 
distance,  /_  ZPM  its  hour  angle. 


§  67.    The  Astronomical  Triangle. 

The  triangle  ZPM  (Figs.  50  and  52)  is  often  called  the 
astronomical  triangle,  on  account  of  its  importance  in  problems 
in  Nautical  Astronomy. 

The  side  PZ  is  equal  to  the  complement  of  the  latitude  of 
the  observer.  For  (Fig.  50)  the  angle  ZOB  between  the  zenith 
of  the  observer  and  the  celestial  equator  is  obviously  equal  to 
his  latitude,  and  the  angle  POZ  is  the  complement  of  ZOB. 
The  arc  NP  being  the  complement  of  PZ,  it  follows  that  the 
altitude  of  the  elevated  pole  is  equal  to  the  latitude  of  the  place 
of  observation. 

The  triangle  ZPM  then  (however  much  it  may  vary  in 
shape  for  different  positions  of  the  star  M)  always  contains 
the  following  five  magnitudes  : 

PZ  =  co-latitude  of  observer  =  90°  —  I, 

ZM=  zenith  distance  of  star  =  z, 

PZM=  azimuth  of  star  =  a, 

PM=^  polar  distance  of  star   =p, 

ZPM=  hour  angle  of  star         =  t. 


APPLICATIONS.  177 

A  very  simple  relation  exists  between  the  hour  angle  of  the 
sun  and  the  local  (apparent)  time  of  day.  Since  the  hourly 
rate  at  which  the  sun  appears  to  move  from  east  to  west  is 
15°,  and  it  is  apparent  noon  when  the  sun  is  on  the  meridian 
of  a  place,  it  is  evident  that  if  hour  angle  =  0°,  15°,  — 15°,  etc., 
time  of  day  is  noon,  1  o'clock  p.m.,  11  o'clock  a.m.,  etc. 

In  general,  if  t  denotes  the  absolute  value  of  the  hour  angle, 

time  of  day  =  —  p.m.,  or  12  —  3-3  a.m., 
according  as  the  sun  is  west  or  east  of  the  meridian. 

§  68.    Problem. 

Given  the  latitude  of  the  observer  and  the  altitude  and  the 
azimuth  of  a  star,  to  find  its  decimation  and  its  hour  angle. 

In  the  triangle  ZPM  (Fig.  52), 
given  p^  ==  90°  —  Z  =  co-latitude, 

ZM=  90°  —  h  =  co-altitude, 
Z.  PZM  =^a  .  =  azimuth, 

to  find  FM=  90°  — d  =  polar  distance, 

Z  ZFM  =t  =  hour  angle. 

Draw  MQ  ±  to  JSfS,  and  put  ZQ  =  m, 

then,  if  a<  90°,  FQ  =  90°  -  (Z  -f  m), 

and  if  a> 90°,  FQ  =  90° - (Z- m)  ; 

and,  by  Napier's  Rules, 

cos  a  =  =h  tan  m  tan  h, 
sin  d  =  cos  FQ  cos  MQ, 
sin  h  =  cos  m  cos  MQ ; 
whence,  tanm=  ±  cot  h  cos  a, 

sin  d  =  sin  h  sin  (I  ±  m)  sec  m,    ^^ — 

in  which  the  —  sign  is  to  be  used  if  a  >  90°.     The  hour  angle 
may  then  be  found  by  means  of  [44],  whence  we  have 
sin  t  =  sin  a  cos  h  sec  d. 


-^- 


178 


SPHERICAL    TRIGONOMETRY. 


§  69.    Problem. 

To  find  the  hour  angle  of  a  heavenly  body  when  its  declina-  ^ 

tion,  its  altitude,  and  the  lati- 
tude of  the  place  are  known. 

In  the  triangle  ZPM  (Fig. 
53), 

given         PZ  =  <dO°  —  l, 

FM=90°-d=p, 
ZM  =  90°  — h; 

required 

ZZPM=t. 

If,  in   the   first  formula  of 
[47], 


sin  ^^  =  Vsin  (s  —  b)  sin  (s  —  c)  esc  b  esc  c, 
we  put 

A=:^t,       a  =  90°-A,       b=p,       c  =  90°  —  l, 
we  have 

s-b  =  90°-i{l-\-p-^h),       s-c  =  ^(l^p-h), 
and  the  formula  becomes 

sin  I ?f  =  d=  [cos i(l-Yp-{-h)shi^{l-{-p  —  h)  sec  I csc^]^, 

in  which  the  —  sign  is  to  be  taken  when  the  body  is  east  of 
the  meridian. 


If  the  body  is  the  sun,  how  can  the  local  time  be  found 
when  the  hour  angle  has  been  computed  ?     (See  §  67.) 


applications.  179 

§  70.   Problem. 

To  find  the  altitude  and  the  azimuth  of  a  celestial  body,  when 
its  declination,  its  hour  angle,  and  the  latitude  of  the  place  are 
known. 

In  the  triangle  ZPM  (Fig.  53), 
given  PZ  =  90°  — Z, 

PM=^Q°  —  d=p, 
AZPM=t', 
required  ZM=  90°  —  h, 

Z.PZM=a. 

Here  there  are  given  two  sides  and  the  included  angle. 
Placing  PQ^=m,  and  proceeding  as  in  §  68,  we  obtain 

tan  m  =  cot  d  cos  t, 

sin  h  =  sin  (I  -\-  m)  sin  d  sec  m, 

tan  a  =  sec  (I  -{-  m)  tan  t  sin  m, 

in  the  last  of  which  formulas  a  must  be  marked  E.  or  W.,  to 
agree  with  the  hour  angle.  " 

§  71.   Problem. 

To  find  the  latitude  of  the  place  when  the  altitude  of  a  celes- 
tial body,  its  declination,  and  its  hour  angle  are  known. 

In  the  triangle  ZPM  (Fig.  53), 
given  ZM=90°-h, 

PM=90°  —  d, 
ZZPM=f', 
required  PZ  =  90° -I. 

Let  PQz=zm,  ZQ  =  n. 


180 


SPHERICAL    TRIGONOMETRY. 


Then,  by  Napier's  Eules, 


cos  t  =  tan  m  tan  d, 
sin  h  =  cos  n  cos  MQ, 
sin  d  =  cos  m  cos  MQ ; 

whence, 

tan m  =  cote? cos  t, 

cos  7^  =  cos  m  sin  A  esc  dy 

and  it  is  evident  from  the  figure 
that 

in  which  the  sign  +  ov  the  sign 
—  is  to  be  taken  according  as 
the  body  and  the  elevated  pole  are  on  the  same  side  of  the 
prime  vertical  or  on  opposite  sides. 

In  fact,  both  values  of  I  may  be  possible  for  the  same  alti- 
tude and  hour  angle ;  but,  unless  n  is  very  small,  the  two 
values  will  differ  largely  from  each  other,  so  that  the  observer 
has  no  difficulty  in  deciding  which  of  them  should  be  taken. 


§  72.   Problem. 

Gwen  the  declination,  the  right  ascension  of  a  star,  and  the 
obliquity  of  the  ecliptic,  to  find  the 
latitude  a7id  the  longitude  of  the 
star. 

Let  M  (Fig.  55)  be  the  star,  F  be 
the  pole  of  the  equinoctial,  and  Q 
the  pole  of  the  ecliptic. 

Then,  in  the  triangle  PMQ, 
given     P()=e  =  23°27', 
PM=^0°  —  d, 
ZMFQ  =  90°  +  r  (see  Fig.  51 ); 
and  Z  FQM=90°  —  v  (see  Fig.  51). 


Fig.  55. 


required  QM  =  90°  — 


APPLICATIONS.  181 

In  this  case,  also,  two  sides  and  the  included  angle  are 
given.  Draw  MH  \_  to  PQ,  and  meeting  it  produced  at  H^ 
and  let  PH=^  n. 

By  Napier's  Rules, 

sin  r  =  tan  7i  tan  d, 
sin  u  =  cos  (e  -\-  n)  cos  MHy 
sin  d  =  cos  7i  cos  MH, 
sin  (^d-\-n)  =  tan  v  tan  Jf^, 
sin  n  =  tan  r  tan  MH ; 
whence,  tan  n  =  cot  c?  sin  r, 

sin  ■z^  =  sin  d  cos  (e  +  ?z)  sec  n, 
tan  V  =  tan  r  sin  (e  +  n)  esc  ti. 

To  avoid  obtaining  u  from  its  sine  we  may  proceed  as 
follows  : 

From  the  last  two  equations  we  have,  by  division, 

sin  u  =  tan  v  cot  (e  +  ^)  sin  d  cot  r  tan  71. 

By  taking  3III  as  middle  part,  successively,  in  the  triangles 
MQH  and  MFII,  we  obtain 

cos  u  COS  V  =  cos  d  cos  r ; 
whence,  cos  u  =  sec  v  cos  tZ  cos  r. 

From  these  values  of  sin  u  and  cos  u  we  obtain,  by  division, 

tan  u  =  sin  v  cot  (e  +  ^')  tan  c?  esc  r  tan  ti. 
From  the  relation 

sin  r  =  tan  n  tan  c?, 
it  follows  that    tan  d  esc  r  tan  n  =  l. 

Therefore        tan  ^^  =  sin  v  cot  (e  +  n), 
a  formula  by  which  u  can  be  easily  found  after  v  has  been 
computed. 


k- 


182  spherical  trigonometry. 

Exercise  XLII. 

1.  Find  the  dihedral  angle  made  by  adjacent  lateral  faces 
of  a  regular  ten-sided  pyramid ;  given  the  angle  V=  18°,  made 
at  the  vertex  by  two  adjacent  lateral  edges. 

2.  Through  the  foot  of  a  rod  which  makes  the  angle  A  with 
a  plane,  a  straight  line  is  drawn  in  the  plane.  This  line  makes 
the  angle  B  with  the  projection  of  the  rod  upon  the  plane. 
What  angle  does  this  line  make  with  the  rod? 

3.  Find  the  volume  V  of  an  oblique  parallelopipedon ; 
given  the  three  unequal  edges  a,  b,  c,  and  the  three  angles 
I,  m,  n,  which  the  edges  make  with  one  another. 

4.  The  continent  of  Asia  has  nearly  the  shape  of  an 
equilateral  triangle,  the  vertices  being  the  East  Cape,  Cape 
Romania,  and  the  Promontory  of  Baba.  Assuming  each  side 
of  this  triangle  to  be  4800  geographical  miles,  and  the  earth's 
radius  to  be  3440  geographical  miles,  find  the  area  of  the 
triangle  :  (i.)  regarded  as  a  plane  triangle ;  (ii.)  regarded  as  a 
spherical  triangle. 

5.  A  ship  sails  from  a  harbor  in  latitude  I,  and  keeps  on 
the  arc  of  a  great  circle.  Her  course  (or  angle  between  the 
direction  in  which  she  sails  and  the  meridian)  at  starting  is  a. 
Find  where  she  will  cross  the  equator,  her  course  at  the 

^^1^  equator,  and  the  distance  she  has  sailed. 

6.  Two  places  have  the  same  latitude  I,  and  their  distance 
^^  X           apart,  measured  on  an  arc  of  a  great  circle,  is  d.     How  much 

^  greater  is  the  arc  of  the  parallel  of  latitude  between  the  places 

than  the  arc  of  the  great  circle?     Compute  the  results  for 
Z  =  45°,  cZ  =  90°. 

7.  The  shortest  distance  d  between  two  places  and  their 
/.              latitudes  I  and  Z'  are  known.     Find  the  difference  between 

%r'   q'         their  longitudes. 


APPLICATIONS.  183 

8.  Given  the  latitudes  and  longitudes  of  three  places  on 
the  earth's  surface,  and  also  the  radius  of  the  earth ;  show 
how  to  find  the  area  of  the  spherical  triangle  formed  by  arcs 
of  great  circles  passing  through  the  places. 

9.  The  distance  between  Paris  and  Berlin  (that  is,  the  arc 
of  a  great  circle  between  these  places)  is  equal  to  472  geo- 
graphical miles.  The  latitude  of  Paris  is  48°  50'  13";  that  of 
Berlin,  52°  30'  16".  When  it  is  noon  at  Paris  what  time  is  it 
at  Berlin? 

Note.  Owing  to  the  apparent  motion  of  the  sun,  the  local  time  over 
the  earth's  surface  at  any  instant  varies  at  the  rate  of  one  hour  for  15°  of 
longitude  ;  and  the  more  easterly  the  place,  the  later  the  local  time. 

10.  The  altitude  of  the  pole  being  45°,  I  see  a  star  on  the^ 
horizon  and  observe  its  azimuth  to  be  45°;   find  its  polar 
distance. 

11.  Given  the  latitude  I  of  the  observer,  and  the  declination 
d  of  the  sun;  find  the  local  time  (apparent  solar  time)  of 
sunrise  and  sunset,  and  also  the  azimuth  of  the  sun  at  these 
times  (refraction  being  neglected).  When  and  where  does  the 
sun  rise  on  the  longest  day  of  the  year  (at  which  time  d  = 
+  23°  27')  in  Boston  (Z  =  42°  21'),  and  what  is  the  length  of 
the  day  from  sunrise  to  sunset  ?  Also,  find  when  and  where 
the  sun  rises  in  Boston  on  the  shortest  day  of  the  year  (when 
^  =  —  23°  27'),  and  the  length  of  this  day. 

12.  When  is  the  solution  of  the  problem  in  Example  11 
impossible,  and  for  what  places  is  the  solution  impossible? 

13.  Given  the  latitude  of  a  place  and  the  sun's  declination ; 
find  his  altitude  and  azimuth  at  6  o'clock  a.m.  (neglecting 
refraction).  Compute  the  results  for  the  longest  day  of  the 
year  at  Munich  (/  =  48°  9'). 

14.  How  does  the  altitude  of  the  sun  at  6  a.m.  on  a  given 
day  change  as  we  go  from  the  equator  to  the  pole  ?     At  what 


184  SPHERICAL    TRIGONOMETRY. 

time  of  the  year  is  it  a  maximum  at  a  given  place?     (Given 
sin  h  =  sin  I  sin  d.) 

15.  Given  the  latitude  of  a  place  north  of  the  equator,  and 
the  declination  of  the  sun ;  find  the  time  of  day  when  the  sun 
bears  due  east  and  due  west.  Compute  the  results  for  the 
longest  day  at  St.  Petersburg  (I  ==  59°  56'). 

16.  Apply  the  general  result  in  Example  15  (cos  t  =  cotl 
tancZ)  to  the  case  when  the  days  and  nights  are  equal  in 
length  (that  is,  when  d  =  0°).  Why  can  the  sun  in  summer 
never  be  due  east  before  6  a.m.,  or  due  west  after  6  p.m.  ? 
How  does  the  time  of  bearing  due  east  and  due  west  change 
with  the  declination  of  the  sun?  xipply  the  general  result 
to  the  cases  where  l<.  d  and  Z  —  c?.  What  does  it  become  at 
the  north  pole  ? 

17.  Given  the  sun's  declination  and  his  altitude  when  he 
bears  due  east ;  find  the  latitude  of  the  observer.  V''  -     '*'"\   mW^^    "^^ 

18.  At  a  point  0  in  a  horizontal  plane  3IJV  a  staff  OA  is 
fixed,  so  that  its  angle  of  inclination  AOB  with  the  plane  is 
iequal  to  the  latitude  of  the  place,  51°  30'  N.,  and  the  direction 
OB  is  due  north.  What  angle  will  OB  make  with  the  shadow 
of  OA  on  the  plane,  at  1  p.m.,  when  the  sun  is  on  the  equi- 
noctial? 

19.  What  is  the  direction  of  a  wall  in  latitude  52°  30'  N. 
which  casts  no  shadow  at  6  a.m.  on  the  longest  day  of  the  year? 

20.  At  a  certain  place  the  sun  is  observed  to  rise  exactly 
in  the  north-east  point  on  the  longest  day  of  the  year ;  find 
the  latitude  of  the  place. 

21.  Find  the  latitude  of  the  place  at  which  the  sun  sets  at 
10  o'clock  on  the  longest  day. 

22.  To  what  does  the  general  formula  for  the  hour  angle, 
\\    in  §  69,  reduce  when  (i.)  h  =  0°,  (ii.)  1  =  0°  and  d  =  0°,  (iii.) 

lOT  d  =  90°? 


APPLICATIONS.  185 

\^  23.  What  does  the  general  formula  for  the  azimuth  of  a 
celestial  body,  in  §  70,  become  when  t  =  90°  =  6  hours  ? 

yy  24.  Show  that  the  formulas  of  §  71,  if  t  =  90°,  lead  to  the 
equation  sin  Z  =  sin  A- esc  c? ;  and  that  if  c?  =  0°,  they  lead  to 
the  equation  cos  I  =  sin  h  sec  t. 

25.  Given  latitude  of  place  52°  30'  16",  declination  of  star 
38°,  its  hour  angle  28°  17'  15";  find  its  altitude. 

26.  Given  latitude  of  place  51°  19'  20",  polar  distance  of 
star  67°  59'  5",  its  hour  angle  15°  8'  12";  find  its  altitude  and 
its  azimuth. 

27.  Given  the  declination  of  a  star  7°  54',  its  altitude 
22°  45' 12",  its  azimuth  129°  45' 37";  find  its  hour  angle  and 
the  latitude  of  the  observer. 

2S.  Given  the  longitude  v  of  the  sun,  and  the  obliquity  of 
the  ecliptic  6  =  23°  27';  find  the  declination  d,  and  the  right 
ascension  r. 

29.  Given  the  obliquity  of  the  ecliptic  e  =  23°  27',  the 
latitude  of  a  star  51°,  its  longitude  315°;  find  its  declination 
and  its  right  ascension. 

30.  Given  the  latitude  of  place  44°  50'  14",  the  azimuth  of  a 
star  138°  58'  43",  and  its  hour  angle  20°;  find  its  declination. 

31.  Given  latitude  of  place  51°  31'  48",  altitude  of  sun  west 
of  the  meridian  35°  14'  27",  its  declination  +21°  27';  find  the 
local  apparent  time. 

32.  Given  the  latitude  of  a  place  I,  the  polar  distance  p  of 
a  star,  and  its  altitude  A;  find  its  azimuth  a. 


APPEE"DIX. 


FORMULAS. 
Plane  Trigonometry. 

1.    sin^^  +  cos^^  =  1-1 
sin  A 


2.    tsinA 


cos  A 


{sin A  X  csc^  =  l. 
tan^Xcot^  =  l.  ^ 

4.  sin  (x-{-y)^  sin  cc  cos  y  +  cos  x  sin  y. 

5.  cos  (x-\-y)  =  cos  a;  cos  ?/  —  sin  x  sin  ?/. 


6.  tan(a7  +  ?/)  = 

7.  cot  (a;  +  ?/)  = 


tan  ic  4"  t3,n  ?/ 
1  —  tan  X  tan  i/ 

cot  X  cot  2/  — - 1 

cot  ?/  4"  cot  X 


8.  sin  (x  —  y)  =  sin  £c  cos  ?/  —  cos  cc  sin  y. 

9.  cos  (x  —  y)=  cos  ic  cos  ?/  +  sin  a;  sin  y. 

^/^     ,       ,  .        tancc  —  tanv 

10.    tan  (a;  —  ?/)  = -— ^. 

^      1  +  tan  a;  tan  y 


11.    cot  (x--y)  = 


cot  a:  cot  ?/  -}-  1 


cot  2/  —  cot  X 

12.  sin  2  a;  =  2  sin  a;  cos  a;.   "] 

13.  cos  2  a:  =  cos'a;--sin'^a;. 


23. 


§  27. 


§28. 


§29. 


FORMULAS. 


187 


14.  tan2cc  = 

15.  cot2x  = 


2  tan  X 


1  —  tannic 

COt^iC  —  1 


§29. 


—  cos^ 


2cotic 

16.  smiz  =  ±^- 

17.  cosi^  =  ±^— ^2 

H.  r>         .  ,  /l  —  COS  ^ 

18.  tani^  =  ±\/':7-^ 

^  ^  1  +  cos  ^ 

VI  +  cos  ^ 
1 


19.   cot^ 


cos  2; 


§30. 


20.  sin  A  +  sin  ^  ==  2  sin  i(A-\-B)  cos  i(A  —  B). 

21 .  sin  ^  —  sin  ^  =  2  cos  i(A-\-B)  sin  ^  (y1  —  ^). 

22.  cos^  +  cos5  =  2cosi(^  +  5)cosi(^— ^). 

23.  cos.4  — cos^  =  — 2sin^(^+^)sin^(^— ^). 
sin  ^  +  sin  B  _  tan  |- (^  + -^) . 


31. 


24. 
25. 


sin  ^  —  sin  ^      tan  ^(A  —  B) 
a      sin  A 


b       siwB 
26.    a^  =  h'^-\-c^  —  2bcG0sA. 

a  —  h_  tan  ^  (J  —  ^) 
^'    H^~'tani(^+^)* 

28.    sini^=V^^^¥^- 


§33. 
§34. 
§  35. 

§40. 


188 


FORMULAS. 


29. 
30. 

31. 

32. 
33. 
34. 

35. 
36. 
37. 

38. 
39. 

40. 

41. 

42. 
43. 

44. 


cosiA=\^-^ 


^^. 


be 


^  ^       s  s  —  a 


■c) 


s(s  —  a) 


hs-a)(s-b)(s-c)  ^  ^. 
^  s 

tSLU^A  = 


F  = 
F  = 


i  ac  sin  B. 
ahin  B  sin  C 


2sin(^+C) 
i^  =  V^  (5  —  a)  (s  —  b)(s  —  c). 


Spherical  Trigonometry 
cos  c  =  cos  a  cos  b. 
f  sin «;  =  sine  sin^. 
1  sin  ^  =  sin  c  sin  B. 
(  cos  A  =  tan  b  cot  c. 
(  cos  B  =  tan  a  cot  c. 

{cos ^  =  cos  a  sin ^. 
cos^  =  cos  Jsin  J. 
r  sin  &  =  tan  a  cot  A. 
\  sin  <x  =  tan  b  cot  B. 
cos  c  =  cot  ^  cot  ^. 

{sin  0^  sin  B  =  sin ^  sin^. 
sin  a  sin  C  =  sin  c  sin  A. 
sin  J  sin  C  =  sin  c  sin  B. 


§40. 


§41. 


§49. 


§53. 


FORMULAS. 


189 


{COS  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A. 
cos  b  =  cos  a  cos  c  +  sin  a  sin  c  cos  ^. 
cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  (7. 


{cos^  =  — 
cos5  =  — 
cos  (7  =  —  I 


cos^  =  — cos^cos  C  +  sinjg  sin  Ccoso^. 
46.  ^  cos  5  =  —  cos  ^  cos  C  +  sin  A  sin  C  cos  &. 
cos  A  cos  j5  +  sin^  sin  B  cos  c. 


53. 


47. 


48. 


sin  ^  ^  =  Vsin  (s  —  b)  sin  (s  —  c)  esc  5  esc  c. 

cos-|-^  =  Vsin  s  sin  (5  —  a)  esc  b  esc  c. 

tan-J^  =  Vcsc  s  esc  (5  —  a)  sin  (s  —  ^)  sin  (s  —  c). 


sin  ^  a = V—  cos  S  cos  {S — A)  gsc  B  esc  C. 
cos  -J-  a = Vcos  {S —B)cos(S—  C)  esc  B  esc  C 
tan  ^a = V—  cos  ASeos(/S'— ^)sec(>S^— ^)see(/S^—  C) . 


§54. 


49. 


50.  J 


cos  ^  (^  +  5)  cos  ^  c  =  COS  i(a-\-b)  sin  ^  C. 
sin K^  +  ^)  cos ^ c  =  cos i(a—b) cos |  (7. 
COS  i(A  —  B)smic  =  sin  ^  (a  +  ^')  sin  |  C. 
sin  -J-  (^  —  ^)  sin  -J-c  =  sin  ^  (a  —  J)  cos -J  C, 

^  ^       COS  -J  (a  -f  ^>)         ^ 


tan  ^  («  +  b) 


Gosi(A—B) 
cosi(A-\-B) 


tan-J-c. 


1/        7N         sin4-M— ^)  ^ 


55. 


51.   F 


■R^E 

180  * 


52.     tan^|-^=tan-|-5  tan-J-(s — <z)tan-|-(s— ^)tan-|-(5— -c). 


y%^2. 


190  FORMULAS. 

Prof.  Blakslee's  construction  by  which  the  direction  ratios 
for  plane  right  triangles  give  directly  from  a  figure  the  analo- 
gies for  a  right  trihedral  or  for  a  right  spherical  triangle: 


tan  b 


The  construction  consists  of  two  parts. 

(a)  Lay  off  from  the  vertex  V  a  unit's  distance  on  each  edge. 

(b)  Pass  through  the  three  extremities  of  these  distances 
three  planes  perpendicular  to  one  of  the  edges,  as  VA.  Now 
these  three  parallel  planes  will  cut  out  three  similar  right 
triangles.  The  first  being  constructed  in  either  of  the  two 
usual  ways,  the  construction  of  the  others  is  evident. 

Since  the  plane  angles  Ai,  A^,  A^  all  equal  the  dihedral  A, 
and  the  nine  right  triangles  in  the  three  faces  give  the  values 
in  the  figure,  we  have  : 

(1)  sin  A  =  sin  a  :  sin  h ;  similarly,  sin  B  =  sin  b  :  sin  h. 

(2)  cos  A  =  tan  b  :  tan  h ;  similarly,  cos  B  =  tan  a  :  tan  h. 

(3)  tan  A  =  tan  a  :  sin  ^ ;  similarly,  tan  B  =  tan  b  ;  sin  a. 

(4)  cos  h  =  Gos  a  cos  b ;  (by  3)  =  cot  A  cot  B. 

(5)  sin^  =  cos -5  :  cos  b  ;  sin  B   =  cos  A  :  cos  a. 

Note.  If  a  sphere  of  unit  radius  be  described  about  F  as  a  centre, 
the  three  faces  will  cut  out  a  right  spherical  triangle,  having  the  sides  a, 
6,  and  h,  and  angles  A,  B,  and  H.  The  above  formulas  are  thus  seen  to 
be  the  analogies  of  : 


FORMULAS.  191 

(1)  sin  A  =  a:  h;  sin  B  =  b  :  h. 

(2)  cosA=  h:  h;  cosB  =  a:  h. 

(3)  tan^  =  a:b;  tan  B  =  b  :  a. 

(4)  h^  =  a^  +  b^;  1  =  sin2  +  cos^ ;  1  =  cot  ^  cot  B. 

(5)  sin  ^  =  cos  JB ;  sin  5  =  cos  A. 

Napier's  rules  give  only  the  following,  whichi  follow  from  the  analogies 
numbered : 


By  (  sin  a  =  sin  Asinh=  tan  b  cot  B  ) 
ID   (  sin  6  =  sin  ^  sin  ^  =  tanacot  J.  ) 


(1)   (  sin  b 

^  '   \  cos  B=  sin  A  cos  b  =  tan 

(4)  J  cos  ^  =  cos  a  cos  6  =  cot^  cot  JB  ]  (4) 


cos  J.=  sin  5  cos  a  =  tan  6  cot  ^  ) 
a  cot  h)  ^ 


The  Gauss  Equations. 


cos  i(A-\-B)GOsic  =  cos  i(a-^b)  sin  -J-  C. 
sin  i(A-{-B)cosic  =  cos  ^(a  —  b)  cos  i  C. 
cos  ^{A  —  B)^ui^c  =  sin ^{a-\-h)  sin ^  C. 
sin  ^  (^  —  ^)  sin  -J-  c  =  sin  ^{a  —  h)  cos  -J-  C. 
fi(A±B)Xf^c=    fi(a±b)XfiC. 

Rule  I.     sin  in  (I.)  gives  —  in  (3),  and  conversely, 
cos  in  (I.)  gives  +  in  (3),  and  conversely. 

Rule  II.   Functions  have  same  names  in  (2)  and  (3). 
Functions  have  co-names  in  (4)  and  (1). 


AN"SWEES. 

PLANE   TRIGONOMETRY. 
Exercise  I. 

o  4  D  i^  lb 

123°  45'  =  ^,   37°30'  =  |^- 
lb  24 

2.  ^-|^=120%  ^=1350,  ^  =  112O30^  ^=168°  45',  ^-84°. 

3.  1°  =  0.0174533  radian.     1' =  0.00029089  radian. 

4.  1  radian  =  206,265''.     7.    14°  27'  28".  10.   3  hr.  49  min.  11  sec. 

g    37f        5^^  8.   69.167  miles.  11.   9  ft.  2  in. 

4  b 

6.    11°  27' -33".  9.   57  ft.  3.55  in.  12.   ^l^sec. 


Exercise  II. 

1.   sin  JB  =  - ,  cos  B  =  -  1  tan  B  =  -i  cot  B= -^  secB  =  - ■>  esc  B  = -' 
c  c  a  0  a  b 

3.  (i.)  sin  =  I,  cos=  |,  (ii.)  sin  =  y\,  etc.         (v.)  sin  =  ff,  etc. 

tan  =  f,  cot  =  I,  (iii.)  sin  =  yV»  ^tc.       (vi.)  sin  =  i^f ,  etc. 

sec  =  f,  CSC  =  f,  (iv.)  sin  =  j\,  etc. 

4.  The  required  condition  is  that  a^  +  6^  =  c^.     It  is. 

5.  (1.)  sin=-T— — 5,  etc.  (m.)  sin  =  -,  etc. 

(ii.)  sm  =    „  ,  % '  etc.  (iv.)  sm  =  —  >  etc. 

7.   In  (iii.)  pV  -j-  ^232  =  ^252 .    in  (iv.)  m^n^s^  +  m^H'^  =  n^^r^. 
S.   c=  145 ;         whence,  sin  A  =  j\%  =cosB;  cos  ^  =  i-ff  =  sin  B ; 
tan  A  =  jW  =  cot  5 ;  cot  A  =  Vi^-  =  tan  5 ;  sec  A  =  f||  =  esc  ^ ;  etc. 


TRIGONOMETKY. 


9.    b  =  0.023  ;  whence,  tan  A  =  cotB  =  ^H- ;   cot  ^  =  tan  B  =  ^%\,  etc. 
10.    a=  16.8  ;  whence,  sin  A  =  J  ||  =  cos  B,  etc. 


11.  c  =  ■»  +  g  ;  whence,  sin  ^  =  ^ — ^  =  cos  5 ;  etc. 

12.  b='^q{p  +  q);  whence,  tan  A  =  -l/-  =  cot  E ;  etc. 

P  —  Q 

13.  a  =  p  —  q  :  whence,  sin^=  — r-^  =  cos  B ;  etc. 

^      ^'  p+  q 

14.  sin  ^  =  2  V5  =  0.89443  ;  etc.  15.    sin  vl  =  | ;  etc. 

16.  sin  J.  =  }  (5  +  V?)  =  0.95572  ;  etc. 

17.  cos^=^(V31- 1)^0.57097;  sin^  =  i  (V31  +  1)  =  0.82097;  etc. 

18.  a  =12.3.  20.    a  =9.                            22.    c  =  40. 

19.  6=1.54.  21.    6  =  68.                           23.    c=  229.62. 
24.  Construct  a  rt.  A  with  legs  equal  to  3  and  2  respectively ;   then 

construct  a  similar  A  with  hypotenuse  equal  to  6. 
In  like  manner,  25,  26,  27,  may  be  solved. 
28.    a  =  1.5  miles ;  6  =  2  miles.  31.   400,000  miles. 

30.   a  =  0.342,  6  =  0.940;  a  =  1.368,  6=  3.760.  32.    142.926  yards. 


Exercise    III. 

5.  Through  A  (Fig.  3)  draw  a  tangent,  and  take  J.  T  =  3  ;  the  angle 

J. or  is  the  required  angle. 

6.  From  0  (  Fig.  3)  as  a  centre,  with  a  radius  =  2,  describe  an  arc  cut- 

ting at  Tthe  tangent  drawn  through  B\  the  angle  ^OT  is  the 
required  angle. 

7.  In  Fig.  3,  take  OM  =  ^,  and  erect  MP  A_  OA  and  intersecting  the 

circumference  at  P ;  the  angle  POM  is  the  required  angle. 

8.  Since  sin  cc  =  cos  x,  OM  =  PM  (Fig.  3),  and  x  =  45°  ;  hence,  con- 

struct X  =  45°. 

9.  Construct  a  rt.  A  with  one  leg  =  twice  the  other  ;  the  angle  opposite 

the  longer  leg  is  the  required  angle. 

10.  Divide  OA  (Fig.  3)  into  four  equal  parts ;  at  the  first  point  of  divi- 
sion from  O  erect  a  perpendicular  to  meet  the  circumference  at 
some  point  P.     Join  OP  ;  the  angle  A  OP  is  the  required  angle. 

21.   r  sin  x.  22.    Leg  adjacent  to  A  =  nc^  leg  opposite  to  ^  =  mc. 


1. 

cos  60°. 

cot  1°. 

sin  45°. 

tan  75°. 

2. 

cos  30°. 

cot  33°. 

sin  15°. 

tan  6°. 

3. 

iV3 

4. 

tan^  = 

cot^ 

=  cot (90° 

5. 

30°. 

7.    90°. 

6. 

30°. 

8.   60°. 

ANSWERS. 


Exercise    IV. 

sec  71°  50'.  tan  7°  41^ 

sin  52°  36'.  sec  35°  14' 

sec  20°  58'.  tan  0°  1'. 

sin  4°  21'.  sec  44°  59' 

A);  hence,  ^  =  90°  -  ^  and  ^  =  45°. 

9.   22°  30'.  11.    10°. 

10.    18°.  90° 

1^. 


n+1 
Exercise  VI. 

cot  ^  =  y\,    sec  A  =  -U,        esc  ^  =  \ |. 

2.  cos^=0.6,  tan  J[  =  1.3333,  cot^  =  0.75,  sec^=1.6667,  csc^=1.25. 

3.  sin  ^=  ^},  tan  J.=  l^i,        cot^=ff,    secJ.=  fi,        cscA  =  ^\. 

4.  sin  J.=0.96,  tan  ^=3.4286,     cot^  =  0.2917,    sec  J.=3.5714. 

5.  sin  J. =0.8,  cos  J. =0.6,        cot  J.=0.75,  sec  J.=l. 6667,  esc ^  =  1.25. 

6.  sin  J.=W2,  cos  J.=i\^,  tan  J.=l,       secJ.=  V2,       cscJ.=  V2. 

7.  tan^=2,        sin  ^=0.90,   cos ^=0.45,  sec  ^=2.22,      esc  ^  =  1.11. 

8.  cos7l=i,       sin  J.=iV3,  tan  J.=  V3,  cot^=iV3,.     cscvl=fV3- 

9.  sin^=|A/2,  cos J.=iV2,  tan^=l,      cot^  =  l,  secJ.=  V2. 

10.  cos  ^=  Vl  —  nfi,  tan^  =  ,    ^   ^Vl  — m^,  cot^=-Vl  — m2. 

'  1  —  m2  m 

,     1  —  m2       ^        ,        2m  ^    .     1  —  m2  .1  +  ^2 

11.  cosJ.  =  — -; -1     tan^=:; -,  cot^=— ,  sec^  = 


1  +  m2  1  —  m2  2m  1  —  rri 

.^      .      ,     m2  — n2      ^        ^     m2  — ri2  m^+rfi 

12.  sm  ^=— T"^ — o-    tan  ^=-- ,sec^=— r 

13.  cot  =  1,  sin  =  1 V2,     cos  =  i  V2,    sec  =  V2,       esc  =  \^. 

14.  cos  =  i  V3,    tan  =  i  Vs,     cot  =  V3,       sec  =  f  Vs,     esc  =  2. 

15.  sin  =  i  V3,    cos  =  i,  tan  =  V3,       cot  =  i  V3,     sec  =  2. 

16.  sin  =  1  V2  -  V3,     cos  =  1  V2  +  V3,     cot  =  2  +  Vs. 

17.  sin  =  I  V2^  ^,     cos  =  1  V2  +  V2,     tan  =  V2  -  1. 

18.  cos  =  1,    tan  =  0,    cot  =  00,    sec=l,    esc  =  00. 

19.  cos  =  0,    tan  =  oo,  cot  =  0,      sec  =  oo,  esc  =  1. 

20.  sin  =  1,    cos  =  0,    cot  =  0,      sec  =  oo,  esc  =  1. 


TRIGONOMETRY. 


21.  cos^  =  Vl-sin2^,   tanA  =  ^r^£=,  esc  A  =  ~ 

22.  sin^ 
sec  A 

23.  sin^- 


Vl  —  sin2  A  '  """  ""      sin  A 

COS  ^  V  1  —  f^ne2  >1 

1 


COS^  ' 

tan  J: 


csc^  = 


cos^ 
1 


Vl  —  cos^  J. 

1 


sec  A 
24.   tan^ 


—     /  = )  cos  ^  =  

VH-  tan^^  Vl  +  tanM 

=  Vl  +  tanM,   cscA  =  ^^^^±^^ 
tSinA 


,  cot  J.  = 


tan^ 


cos^  =: 


cot^ 
cot^ 


CSC  ^  =  Vl  +  cot2^,  sin  A 


sec  A  = 


Vl  +  cot2  A 


Vl  +  cot2^ 


Vl  +  cot2  A  '        ^^  cot  A 

iV5,      cos^  =  f^/^.  27.   smA  =  ^\,    cos^  =  ff. 

1  — .3cos2^  4-8cos4^ 


25.  sin^ 

26.  sin  A  =  ^Vl5,    tan  A  =  Vl5, 


28. 


cos2  A  —  cos*  A 


Exercise  VII. 


1.  a;  =  45°.  5.  x  =  60°. 

2.  x  =  30°.  6.  x  =  45°. 

3.  a;  =  0°,  or  60°.  7.  x  =  45°. 

4.  x  =  45°.  S.  x  =  45°. 


1. 

6 

-  =  cos  vl  ;  .• 

b 

•  C=  7- 

cos^ 

2. 

-  =  sm  ^  ;  .• 

6 

-=  cos  J.;  .• 

r-       '^ 

sin  A 

3. 

.  b  =  c  cos  ^. 

4. 

6. 

b 

-  =  cos  A  :  .• 
c                 ' 

^  =  90°  -  B, 
a  =  ccos  B, 
6  =  c  sin  B. 

6 

•  C=  7- 

COS  J. 

9.   a;  =  60°.         13.   x  =  0°,  or  60°. 

10.   a;  =60^ 

14.   x=30°. 

11.   a;  =  30= 

15.   X  =  30°,  or  45° 

12.   a;  =  45'^ 

16.   a;  =  45°. 

17.   x  =  60°. 

E    VIII. 

6. 

^  =  90°  -  B, 

a  =  bcot  Bf 

b 
sin  B 

7. 

^  =  90°  -  B, 

6  =  a  tan  J5, 

cosB 

8.  cos  ^  =  - » 
c 

B=90°-A, 

a=  ^c^-  62. 

ANSWKRS. 

fi 

Exercise  IX. 

31.    c=  7.8112, 

A  =  39°  48', 

B  =  50°  12', 

F=15. 

32.    6=69.997, 

A  =  30'  12", 

B  =  89°  29'  48", 

F=  21.525. 

33.    a  =1.1886, 

A  =  43°  20', 

B  =  46°  40', 

ji^=  0.74876. 

34.    6  =  21.249, 

c  =  22.372, 

B  =  71°  46', 

i^=  74.372. 

35.    a  =  6.6882, 

c=  13.738, 

B=60°52', 

i?'=  40.129. 

36.   a  =  63.859, 

6=23.369, 

B  =  20°  6', 

i?'=  746.15. 

37.    a  =19.40, 

6  =  18^778, 

A  =  45°  56', 

-F=  182.15. 

38.    6  =  53.719, 

c=  71.377, 

^=41°  11', 

F  =  1262.4. 

39.    a  =12.981, 

c=  15.796, 

A  =  55°  16', 

2?^=  58.416. 

40.   a  =  0.58046, 

6=8.442, 

A=    3°  56', 

i?'=  2.4501. 

41.  F=i{c^  sin  A  cos  A). 

43.    F=i(62tan^). 

42.  F=i{a'^ cot  A). 

44.    F=i( 

;aVc2  —  a2). 

45.    6=11.6,      c  = 

15.315,  A  = 

40°  45' 48",  5  =  49°  14' 12". 

46.   a  =  7.2,        c  = 

8.7658,  B  = 

34°  46'  40"   A  =  55°  13'  20". 

47.   a  =  3.6474,  6  = 

6.58,        c  = 

7.5233,           B  =  6] 

L°. 

48.   a  =10.283,  6  = 

19.449,  A  = 

27°  52',          B  =  62°  8'. 

49.  19°  28'  17"  and  70°  31'  43". 

50.  3  and  5.1961. 

90° 

51.  a  =  c  cos — -— ' 

n  +  1 

57.  59°  44' 35" 

58.  95.34  feet. 
•  59.    1°  25'  56". 

• 

6  =  c  sm  — — r ' 
n+1 

60.  7.0712  miles  in  each  direction. 

61.  20.88  feet. 

52.    36°  52'  12"  and 

53°  7'  48". 

62.   56.65  feet. 

53.   212.1  feet. 

63.   228.63  yards. 

54.    732.22  feet. 

64.    136.6  feet. 

55.    3270  feet. 

65.    140  feet. 

56.   37.3  feet,  96  feet. 

66.   84.74  feet. 

Exercise  X. 

1.  C  =  2    (90°  — J.),     c  =  2acos^,  h=asmA. 

2.  A  =  i{lSO°—  C),     c  =  2acosA,  h=asinA. 

3.  C  =  2   (90°  —  ^),     a  =  c -^2  008^,  h=asmA. 


TRIGONOMETRY. 


4. 

^  =  1(180°- C), 

a  =  c  -^  2  cos  A, 

h=  a  sin  A. 

5. 

C  =  2    (90°-^), 

a  =  h-7-smA, 

c  =  2  a  cos  ^. 

6. 

^  =  i(180°-C'), 

a  =  y^-7-sin^, 

c  =  2  a  cos  ^. 

7. 

sin  J.  =  h-^  a, 

C  =  2(90°-^), 

c  =  2acos^. 

8. 

tan  J.  =  h-7-  ^c, 

(7  =  2(90°-^), 

a  =  /i-^sin^. 

9. 

A  =  67°  22'  50'^ 

C  =  45°  14'  20", 

/i=13.2. 

10. 

c  =  0.21943, 

h  =  0.27384, 

F=  0.03004. 

11. 

a  =  2.055, 

;i=  1.6852, 

F=  1.9819. 

12. 

a  =  7.706, 

c  =  3.6676,       ' 

2^^=13.725. 

13. 

A  =  79°  36'  30'', 

C  =  20°47', 

c  =  2.4206. 

14. 

A  =  77°  19'  11", 

C  =  25°  21'  38", 

a  =  20.5. 

15. 

^  =  25°  27' 47", 

0=129°  4' 26", 

a  =  81.41,          h  =  S5. 

16. 

^  =  81°  12' 9", 

C=  17°  35' 42", 

a  =17,               c  =  5.2. 

17. 

F=icV4a2_c2. 

22.    0.76537. 

18. 

F=  a2sin|Ocos^ 

C. 

23.    94°20'.  . 

19. 

F=  a2sin^cos^. 

24.   2.7261. 

20. 

F=A2tan^C. 

25.   38°  56' 33". 

21. 

28.284  feet,  4525.44  sq.  feet. 

26.   37.699. 

Exercise   XI. 


1. 

r=  1.618, 

;i=  1.5388,     F=  7.694. 

2. 

r=  11.271, 

h=10.SS6,     F=  381.04. 

3. 

h  =  0.9848, 

p  =  6.2514,     F=  3.0782. 

4. 

A  =19.754, 

c  =  6.2572,     F=1236. 

5. 

r=  1.0824, 

c  =  0.82842,  F=  3.3137. 

6. 

r  =  2.5933, 

A  =2.4882,      c=  1.4615. 

7. 

r=  1.5994, 

7^=1.441,       p  =  9.716. 

8. 

0.61803. 

12.   0.2238.             17. 

11.636. 

9. 

0.64984. 

13.    0.31.                18. 

99.64. 

10. 

0.51764. 

14.    0.82842.           19. 

1.0235. 

11. 

5-        c 

15.  94.63.              20. 

16.  414.97. 

0.635. 

o       90° 

ANSWERS. 


Exercise    XII. 

5.  Two  angles  :  one  in  Quadrant  I. ,  the  other  in  Quadrant  II. 

6.  Four  values  :  two  in  Quadrant  I. ,  two  in  Quadrant  IV. 

7.  X  may  have  two  values  in  the  first  case,  and  one  value  in  each  of  the 

other  cases. 

8.  If  cos  cc  =  —  f ,  ic  is  between  90°  and  270°  ;  if  cot  cc  =  4,  x  is  between 

0°  and  90°  or  180°  and  270° ;  if  sec  z  =  SO,x  is  between  0°  and  90° 
or  between  270°  and  360° ;  if  esc  x  =  —  3,  cc  is  between  180°  and 
360°. 

9.  In  Quadrant  III.  ;  in  Quadrant  II.  ;  in  Quadrant  III. 

10.  40  angles  ;  20  positive  and  20  negative. 

11.  +,  when  X  is  known  to  be  in  Quadrant  I.  or  IV. ;  — ,  when  x  is 

known  to  be  in  Quadrant  II.  or  III. 

14.  sinx  =  — fV3,      tanx  =  — 4V3,      cot x  =  —  y^ V3,  esc x  =  —  ^^ V3. 

15.  sinx  =  ±  Jq  VlO,  cosx  =  qi  j^VlO,  tanx  =  — ^,         secx  =  :piVlO. 

cscx  =  ±  VlO. 

16.  The  cosine,  the  tangent,  the  cotangent,  and  the  secant  are  negative 

when  the  angle  is  obtuse. 

17.  Sine  and  cosecant  leave  it  doubtful  whether  the  angle  is  an  acute 

angle  or  an  obtuse  angle  ;  the  other  functions,  if  +  determine  an 
acute  angle,  if  —  an  obtuse  angle. 

20.  sin  450°  =  sin  (860°+  00°) = sin  90°=  1 ;  tan  540°  =  tan  180°  =  0  ; 
cos  630°  =  cos  270°  =  0  ;  cot  720°  =  cot  0°  =  00  ; 
sin  810°  =  sin  90°    =  1  ;  esc  900°  =  esc  180°  =  00. 

21.  46°,  135°,  225°,  315°.  22.    0.  23.    0.  24.    0. 
25.   a2-62  +  4a&. 

Exercise  XIII. 

2.  sin  172°=      sin    8°.  8.  sin  204°  =— sin  24°. 

3.  cos  100°  =  —  sin  10°.  9.  cos  359°  =      cos    1°. 

4.  tan  125°  =  -  cot  35°.  10.  tan  300°  =  -  cot  30°. 

5.  cot    91°  =  —  tan    1°.  11.  cot  264°  =  tan  6°. 

6.  sec  110°  =  —  CSC  20°.  12.  sec  244°  =  -  esc  26°. 

7.  esc  157°  =      CSC  23°.  13.  esc  271°  =  -  sec  1°. 


8  TRIGONOMETRY. 

14.  sin  163°  49'  =       sin  16°  11',  17.    cot  139°  17'  =  -  cot  40°  43'. 

15.  cos  195°  33'  =  -  cos  15°  33'.  18.   sec  299°  45'  =      esc  29°  45'. 

16.  tan  269°  15'  =      cot    0°  45'.  19.    esc    92°  25'  =      sec    2°  25'. 

20.  sin  (—   75°)  =  -  sin   75°  =  -  cos  15°, 
cos(—    75°)=      cos  75°=      sin  15°,  etc. 

21.  sin  (-  127°)  =  -  sin  127°  =  -  cos  37°, 
cos  (-  127°)  =      cos  127°  =  -  sin  37°,  etc. 

22.  sin  (—  200°)  =      sin  160°  =       sin  20°, 
cos(—  200°)  =      cos  200°  =  —  cos  20°,  etc. 

23.  sin  (-  345°)  =  -  sin  345°  =       sin  15°, 
cos  (—  345°)  =      cos  345°  =      cos  15°,  etc. 

24.  sin  (-    52°  37')  =  -  sin    52°  37'  =  -  cos  37°  23', 
cos(-    52°  37')  =      cos  52°  37'  =      sin  37°  23',  etc. 

25.  sin  (-  196°  54')  =  -  sin  196°  54'  =      sin  16°  54', 

cos  (-  196°  54')  =      cos  196°  54'  =  -  cos  16°  54',  etc.  . 

26.  sin  120°  =      i  V3,  cos  120°  =  -  i,  etc. 

27.  sin  135°  =  +  i\^,  cos  135°  =  -  i  V2,  etc. 

28.  sin  150°  =  +  h        cos  150°  =  —  -J  V3,  etc.' 

29.  sin  210°  =  -  i,        cos  210°  =  -  |  V3,  etc. 

30.  sin  225°  =  —  i  ^,  cos  225°  =  —  i  V2,  etc. 

31.  sin  240°  =  -  |  Vs,  cos  240°  =  -  i,  etc. 

32.  sin  300°  =  —  i  V3,  cos  300°  =  +  i,  etc. 

33.  sin  (-  30°)  =  -h  cos  (-  30°)  =  +  ^ Vs,  etc. 

34.  sin  (-  225°)  =  +  i  V2,  cos  (-  225°)  =  -  i^/^,  etc. 

35.  cos  a;  =  —  i  V2  or  —  V^,  etc.,  x  =  225°. 

36.  tan  X  =  —  V^,  sin  x  =  i,  cos  a;  =  —  iV3,  x  =  150°. 

37.  sin  3540°  =  sin  300°  =  -  sin  60°  =  -  ^  V3,  cos  3540°  =+h  etc. 

38.  210°  and  330°;    120°  and  300°. 

39.  135°,  225°,  and  -  225°;   150°  and  -  30°. 

40.  30°,  150°,  390°,  and  510°. 

41.  sin  168°,  cos  334°,  tan  225°,  cot  252°, 
sin  349°,  cos  240°,  tan  64°,  cot  177°. 

42.  0.848.     (Hint :  tan  238°  =  tan  58°,  sin  122°  =  sin  58°.) 

43.  —  1.952.  45.    m  sin  x  cos  x. 

44.  (a  —  6)  sin  x.  46.    (a  —  b)  cot  x—  (a  +  b)  tan  x. 


ANSWEES.  9 

47.  a2  +  62  +  2  a6  cos  x.  49.   cos  xsiny  —  sin  x  cos  y. 

48.  0.  .  50.   tancc. 

61.   Positive  between  x  =  0°  and  x  =  135°,  and  between  x  =  315°  and 
X  =  360° ;  negative  between  x  =  135°  and  x  =  315°. 

52.  Positive  between  x  =  45°  and  x  =  225°;  negative  between  x  =  0°  and 

X  =  45°,  and  between  x  =  225°  and  x  =  360°. 

53.  sin  (x  —   90°)  =  —  cos  x,  cos  (x  —    90°)  =  sin  x,  etc. 

54.  sin  (x  —  180°)  =  —  sin  x,  cos  (x  —  180°)  =  —  cos  x,  etc. 


Exercise  XIV. 

1.  sin  (x  +  y)  =  ff,  cos (x  +  y)  =  |f.        2.   cosy,  siny. 

3.  sin  (  90°  +  y)=      cosy,     cos  (  90°  +  y)  =  —  sin  y,  etc. 

4.  sin  (180°  —  y)=      sin ?/,     cos (180°  —  y)  =  —  cosy,  etc. 

6.  sin  (180°  +  ?/)  =  —  sin y,     cos  (180°  +  ?/)  =  —  cosy,  etc.  ' 

6.  sin  (270°  —  y)  =  —  cosy,     cos  (270°  —  y)  =  —  sin y,  etc. 

7.  sin  (270°  +  y)  =  —  cosy,     cos  (270°  +  y)  =      sin  y,  etc. 

8.  sin  (360°  —  y)  =  —  sin  y,     cos  (360°  —  y)  =      cos  y,  etc. 

9.  sin  (360°  +  y)  =      siny,     cos  (360°  +  y)  =      cosy,  etc. 

10.  sin  (x  —    90°)  =  —  cos  x,     cos  (x  —    90°)  =      sin  x,  etc. 

11.  sin  (x  —  180°)  =  —  sin  X,     cos  (x  —  180°)  =  —  cos  x,  etc. 

12.  sin  (x  —  270°)  =      cos  x,     cos  (x  —  270°)  =  —  sin  x,  etc. 

13.  sin  (  —  y)         =  —  sin  y,    cos  (—  y)         =      cos  y,  etc. 

14.  sin(45°—y)  =  |^V2 (cosy— siny),    cos(45°— y)  =  |V2 (cosy + siny),  etc. 

15.  sin(45°+ y)  =  ^V2 (cos y+ siny),    cos(45°+y)  =  |  V2(cosy— siny),  etc. 

16.  sin(30°+y)  =  i(cosy+ V3siny),    cos(30°  +  y)=^(V3  cosy -siny),  etc. 

17.  sin(60°—y)  =  i(V3 cosy— siny),    cos(60°— ?/)  =  |(cosy+ V3siny),  etc. 

18.  3sinx-4sin3x.  19.   4 cos^x  -  3 cos x.  20.   0.         21.   iV3. 


,,  =  ^i^^8  =  0.10051 ;  cos  I X  =  V^±f^  =  0.99494. 
23.    cos2x=  —  i,  tan2x=  —  V3. 


22.    sini.. 


10  TRIGONOMETRY. 


24.  sin  22i°  =  i  V2  -  V2  =  0.3827,  cos  22i°  =  ^  V2  +  V2  =  0.9239. 
tan22i°=V2-l         =0.4142,  cot22|°  =  V2+l        =2.4142. 

25.  sin  15°    =  |- V2 —  V3  =  0.2588,  cos  15°    =  i  V2  +  VS  =  0.9659. 
tan  15°    =2-V3        =0.2679,  cot  15°    =  2  +  Vs        =3.7321. 

27-33.  The  truth  of  these  equations  is  to  be  established  by  expressing 
the  given  functions  in  terms  of  the  same  function  of  the  same  angle. 
Thus,  in  Example  27, 

sin  2  X  =  2  sin  X  cos  x, 

and  2  tan  x  =  2 '      1  +  tan2x  =  sec^x  =  — w-  • 

cos  X  cos-^x 

By  making  these  substitutions  in  the  given  equation  its  truth  will 
be  evident. 

34.  sin  J.  +  sin  £  +  sin  C  =  sin  J:  +  sin  ^  +  sin  [180  -  {A  +  B)] 

=  sin  A -\-  sin  B -i-  sin  {A  +  B) 

=  2  sin  ^  (^  +  JB)  cos^  (^  -  B)  +  2  sin  i  (^  +  B)  cos  1{A  +  B) 

=  2  sin  ^  (^  +  B)  [cos  l{A-B)-{-  cos  i  {A  +  B)] 

=  4sm^(^  +  ^)   cos  ^  J.  cos  I  ^,  (see  §§  29  and  31). 

But  cos^  C  =  cos  [90° -i(^  +  E)]=  sin  1(^  +  5). 

Therefore,  sin  -4  +  sin  jK  +  sin  C  =  4  cos  ^Acos^B  cos  |  C. 

35.  Proof  similar  to  that  for  34. 

__    ^       .   ,  ^      „  .   ,       ^  sin  ^cosg  ,  cos  J.  sin  B  ,  sin  C 

36.  tan  A  +  tan  5  +  tan  C  = :; -\ =:  -\ 

cosJLcoSjB      cos  a  cos  5      cosC 

sin  G       ,  sin  C  sin  C  cos  C  +  cos  A  cos  B  sin  C 


cos  A  cos  5     cos  G  cos  A  cos  jB  cos  (7 

_  (cos  A  cos  B  +  cos  G)  sin  C  _  [cos  AcosB—  cos  {A -h  B)]  sin  G 

~         cos  J.  cos  JS  cos  C7         ~  cos  A  cos  5  cos  C 

sin  A  sin  B  sin  C  ^        .  ,       ^ ,       „ 

= —  =  tan  A  tan  B  tan  C. 

cos  A  cos  B  cos  G 

37.   Proof  similar  to  that  for  36. 

38        2  42.   tan2x.  ^^    cos  (x  +  y) 

sin2x  cos  (x  -  y)  '    sinxsiny 

39.   2  cot 2  X.  '    cos X cosy  47.    tan x  tan y. 

^^^   cos  jx-y)  ^^    cos  (x  +  y) 

sin  X  cos  2/  '    cos  X  cosy 

cos  (x  +  y)  .-     cos  (x  —  y) 

sin  X  cosy  '     sin  x  sin  y 


ANSWERS.  11 


Exercise  XV. 
1.  sin-iiV3  =60°  +  2n7t  or  120°  +  2 n tt. 

tan-i— =  =S0°  +  2n7i:  or  210°  +  2n7t. 
V3 

vers-i  i     =±60°  +  2n7t. 
cos-i /- ^\  =  135°  +  2 n TT  or  225°  +  2n7t. 

csc-i  V2    =  45°  +  2  n  TT  or  135°  +  2  w  ;r. 
tan-i oo      =  90°  +  2 n ;r  or  270°  +2 nit. 
sec-12       =±60°  +  2n^. 
cos-i  (-  ^  VS)  =  150°  +  2  w  ^  or  210°  +  2  n  tt. 

4.    -^.  10.    ±-|.  12.    ±iV2. 

2V2  13 

8.   0°,  90°,  180°.  11.    ±  ^-  13.   «  =  0,  or  ±  |  V3. 


Exercise    XVI. 

1.   If,  for  instance,  B  =  90°,  [25]  becomes  -  =  sin  A. 

3.   a2  =  62  +  c2,        a2  =  62  +  c2-26c,        a"^  =  l^  +  c^  +  2hc.   ' 

6.  90°. 

7.  (i.)  =  tan  (J.  —  45°),  and  a  right  triangle. 

(ii.)  a  +  h=  (a  —  &)  (2  +  V3),  an  isosceles  triangle  with  the  angles 
30°,  30°,  120°. 

Exercise  XVII. 

9.    300  yards.  15.   a  =  5,  c  =  9.6592. 

10.  ^^=59.564  miles.  16.   a  =  7,  6=8.573. 

AC  =  54.285  miles.  17.  sides,  600  feet  and  10-39.2  feet ; 

11.  4.6064  miles,  4.4494  miles,  altitude,  519.6  feet. 
3.7733  miles.  is.  855:1607. 

12.  4.1501  and  8.67.  19.  5.438  and  6.857. 

13.  6.1433  miles  and  8.7918  miles.  20.  15.588. 

14.  8  and  5.4723. 


12 


TRIGONOMETRY. 


11.    420. 


Exercise   XVIII. 

12.    124.617. 


11.  6. 

12.  10.392. 
14.    8.9212. 


Exercise   XIX. 


15.  25. 

16.  3800  yards. 

17.  729.67  yards. 


18.  10.266  miles. 

19.  6.0032  and  2.3385. 

20.  26°  0'  10''  and  14°  5'  50' 


Exercise   XX. 

11.  A  =  36°  52'  12",     B  =  53°  7'  48",     C  =  90°.         16.   45°,  60°,  75°. 

12.  ^=5=.3.3°33'27",  C  =  112°53'6".    17.   4°23' W.  of  N.,  or  W.  of  S. 

13.  A  =  B=C  =  60''.  18.   60°. 

14.  Impossible.  20.   0.88877. 

15.  45°,  120°,  15°.  21.    54.516  miles. 


Exercise  XXI. 


1.  4333600. 

2.  365.68. 

3.  13260. 

4.  8160. 
6.   240. 

6.  26208. 

7.  15540. 

8.  29450  or  6982.8. 


9.   17.3204. 

10.  10.3919. 

11.  0.19975. 

12.  db  sin  A. 

13.  \{a'^-b'^)t&nA. 

14.  2421000. 

16.   30°,  30°,  120°. 


1.  21.166  miles 

2.  6.3399  miles. 

3.  119.29  feet. 

4.  30°. 


Exercise  XXII. 
24.966  miles. 


6.  20  feet. 

6.  2.6247  or  21.4587 

7.  276.14  yards. 

8.  383.35  yards. 


ANSWERS. 


13 


MISCELLANEOUS  EXAMPLES. 


2. 

106.70  feet; 

21. 

260.21  feet; 

46. 

294.69  feet. 

142.86  feet. 

3690.3  feet. 

47. 

12,492.6  feet. 

3. 

1023.9  feet. 

22. 

1.3438  miles. 

48. 

6.3.397  miles. 

4. 

37°  34'  5^'. 

23. 

235.80  yards. 

49. 

210.44  feet. 

5. 

238,400  miles. 

27. 

8  inches. 

51. 

757.50  feet. 

6. 

861,880  miles. 

30. 

460.46  feet. 

52. 

520.01  yards. 

7. 

2922.4  miles. 

31. 

88.936  feet. 

53. 

1366.4  feet. 

8. 

60°. 

32. 

13.657  miles. 

54. 

658.36  pounds; 

9. 

3.2068. 

34. 

56.564  feet. 

22°  23'  47"  with  first 

10. 

0.6031. 

35. 

51.595  feet. 

force. 

11. 

199.56  feet. 

36. 

101.892  feet. 

55. 

88.326  pounds; 

12. 

43.107  feet. 

38. 

N.  76°  56'  E. ; 

45°  37'  16"  with 

13. 

45  feet. 

13.938  miles  an  hr. 

known  force. 

14. 

26°  34'. 

39. 

442.11  yards. 

58. 

500.16;  536.27. 

15. 

78.367  feet. 

40. 

255.78  feet. 

59. 

345.48  feet. 

16. 

75  feet. 

41. 

3121.1  feet ; 

60. 

345.46  yards. 

17. 

1.4446  miles. 

3633.5  feet. 

61. 

61.23  feet. 

18. 

7912.4  miles. 

42. 

529.49  feet. 

63. 

307.77  yards. 

19. 

56.649  feet. 

43. 

41.411  feet. 

64. 

19.8;  35.7;  44.5. 

20. 

69.282  feet. 

44. 

234.51  feet. 

65. 

±45°,  ±136°. 

66. 

45. 

cos  A  ■ 

25.433  miles. 

—  m±  Vw2  +  4  ( 

2 

(n  + 

E. 

67.    sm^  =\h, z 

.    \  1  —  n^ 


cos  B 


_  n     /l  —  m' 
~  m  \  1  -  7i2 


68. 
69. 

±60°,  ±120°.              ^2_   r  =  h^c'''\           B: 
0°,  180°,  ±  60°.                           2          n 

a       180° 
=  2^"*    n 

70. 

0°,  30°,  180°,  210°.       73.    i6csin^. 

74.  i  c2  sin  A  sin  B  esc  {A -\- B). 

75.  \/s{s-a){s-b){s-c). 

14  TRIGONOMETRY. 

77.  199  A.  3  R.  8.  p.  94.    16,281.  114.    S.  56°  T  30''  E.; 

78.  210  A.  3  R.  26  p.         95.    435.76  sq.  ft.  202.6  miles. 

79.  12  A.  3  R.  36  p.  96.    49,088  sq.  ft.         115.    N.  17°  25'  W.; 

80.  3  A.  0  R.  6  p.  97.    749.95  sq.  ft.  37°  46'  N. 

81.  12  a.  1r.  15  p.  98.    422.38  sq.  ft.  116.    S.  56°  11' E.;  244.3. 

82.  4  A.  2  R.  26  p.  99.    1834.95  sq.  ft.       117.    359.87  miles. 

83.  14  A.  2  R.  9  p.  100.    26.87.  121.    Long.  68°  55'  W. 

84.  61  A.  2  R.  103.   6.  122.    103.6  mUes. 

85.  4  A.  2  R.  26  p.  108.   6.  124.    33°  18'  K  ; 

86.  13.93,  23.21,  110.    6086.4  feet.  36°  24'  W. 
32.50  ch.                    HI.    5°  25'  S. ;                125.    N.  28°  47'  E. ; 

87.  9  A.  0  R.  1  p.  457.5  miles.  1293  miles. 

89.  876.34.  112.    460.8  miles;  126.    S.  50°40'W.; 

90.  1229.5.  383.1  miles.  250.8  ;  20°  9' W. 

92.  1076.3.  113.    229  miles;  127.    38°21'N.; 

93.  2660.4.  lat.  11°  39'  S.  55°  12'  W. 
128.  171  miles  ;  32°  44'  W.                129.    N.  36°  52'  W.  ;  36°  8'  W. 

130.  173  miles  ;  51°  16'  S.  ;  34°  13'  E. 

131.  S.  50°  58'  E.  ;  47°  15'  N.  ;  20°  49'  W. 

132.  N.  53°  20'  E.,  16°  7'  W.  ;  or  N.  53°  20'  W.,  25°  53'  W. 

133.  N.  47° 42.5' E.,  19° 27' N.,  121° 51' E.  ;  or  N.  47° 42.5' W.,  19°  27' N., 
116°  9'  E.  ;  or  S.  47°  42. 5' E.,  14°  33' N.,  121°  48'  E.  ;  or  S.  47°  42.5' 
W.,  14°33'N.,  116°  12' E. 

134.  Lat.  30°,  359.82  miles  ;  lat.  45°,  359.73  miles  ;  lat.  60°,  359.50  miles. 

137.  N.  72°  33'  E.  ;  45  miles  ;  42°  15'  N.,  69°  5'  W. 

138.  N.  72°  4'  W.,  287  miles  ;  32°  54'  S.,  13°  2'  E. 


ANSWERS.  15 

PROBLEMS  IN  GONIOMETRY. 

[The  solutions  here  given  are  for  angles  less  than  360°.] 


79    ±J_,±J^.  102.  x=±i7t,  ±^7e. 

V5         V5  •  103.  x  =  0°,  ±60°. 

80.  ±  V5  —  2.  104.  X  =  tan-i  Vi 

81.  ±iV3.  105.  x=-15°,  105°. 

82.  ±1,  ±  f .  106.  X  =  —  2  cot-i  a. 

83.  ±W2-.  ^,^^  x^cos-^r'^^t^^^'V 

84.  h                  _  V  4  y 
or  '"V5-l     V5  +  1  108.  x  =  -45°,  135°, 

^^-         4       '        4       *  isin-i(l-a). 

86.  x=i7t,  f  7t.  109.  x=  ±  30°,     ±  60°,     ±  120°, 

87.  x  =  90°,  270°.  ±150°. 

,  VS  -  1  110.  X  =  ±  60°,  ±  90°,  ±  120°. 

88.  X  =  sm-i  — 

2  111.  x=±60°,  ±90°,  ±120°. 

89.  x  =  0°,  90°.  112.  x=120°. 

90.  x  =  30°,  sin-i(-i).  ii3_  3^  =  300,  150°,  gin-ij. 

91.  X  =  180°,  cos-if.  114.  X  =  ±  60°,  ±  90°. 

92.  x  =  0°,  120°,  180°,  240°.  ng^  ^^0°,  ±20°,  ±100°,  ±140°, 

93.  x  =  45°,  225°,  tan-i(-i).  180°. 

94.  x  =  0°,  ±60°,  ±120°,  180°.      116.  x  =  ±  45°,  ±90°,  ±1.35°. 

95.  x=  - 45°,  135°,  117.  x  =  ±  30°,      ±  60°,      ±  90°, 
i  sin-i  (2  V2  -  2).  ±120°,  ±150°. 

96.  X  =  0°,  45°,  180°,  225°.  118.  x  =  0°,  45°,  ±  90°,  225°. 

.         I~Y\  110.  X  =  ±  30°,     ±  60°,     ±  120°, 

97.  x  =  coB-^i^±yj-^y  ±1500. 

96.    x  =  0°,    45°,  90°,   180°,   225°,    120.  x  =  ±  30°,  ±90°,  ±150°. 

270°.  121.  x  =  0°,  45°,  180°,  225°. 

99.    x  =  0°,  180°,  isin-if.  122.  x  =  ±  45°,     ±60°,     ±120°, 

100.  x  =  0°,  ±90°,  ±120°.  ±135°. 

101.  x=0°,  ±36°,  ±72°,  ±108°,    123.  x  =  0°,  ±45°,  ±135°. 

±  144°,  180°.  124.  X  =  ±  30°,  ±  90°,  ±  150°. 


16 


TRIGONOMETRY. 


125.  X  =  8°,  168°.  139.   x  ==  ±  |. 

126.  aj  =  tan-iVf  140.   x  =  l. 

127.  x  =  ±  30°.  141.   x  =  0,  1,  -  1. 

128.  x=±  60°,  ±  120°  142.    x  =  ±  V|. 

129.  x=±30°,    ±60°,    ±120°,  _    1 
±150°.  ^^^'   ^~'^' 

130.  x=±sin-i|.  144^    (a^  +  65)1. 

131.  X  =  30°,  150°  -  cos-i ^  •  / 1  ±  m\^ 

«/^  145.    ('-f^y(lT2m). 

132.  x^tan-i^^^,  -tan-if.  ^^g    ^ 

133.  2/  =  —  90°,  X  indeterminate  ;  . .«  ,  . /t  ■  ,  ,/o 
x  =  45°,  y  =  0°;  ■^'  =^^^'^- 
x=135°,  y=:180°;  ^^^'    ^'  ~t- 

x  =  225°,  y  =  0°',  149       «+l    , 
x  =  315°,  2/ =180°.  *    V2a+1 

134.  X  -  tani  ^6 '  151.   tan  (x  +  y). 

y  =  t^n-i^^±:^/^^ZM,  152.   '4^- 
2  6  sm  y 

135.  x  =  45°,  225°.  153.   -tanx. 

136.  x=±l,  ±V3.  ^^^    tan--^. 

138.   x  =  |V3.  156.   cot2a:-tan2x. 


ENTRANCE    EXAMINATION   PAPERS. 

I. 

a         .      90°  90° 

6.   rsm^j-q-j'       '•cos^i^p^-  7.   475.27  feet. 

n. 

4.   sin  =  i  V2-V2,  tan  =  V2  -  1,         sec  =  V4-2V2, 

cos  =  iV2+V2,  cot  =  V2+l,         csc=  V4  +  2V2. 

6.  (i.)  one,     (ii.)  none,  (iii.)  none,     (iv.)  two. 

7.  383.36  yards. 


ANSWERS.  17 


2.    (a)sin^=±i,        tan^^rp     /  '     cot^=qiV3, 


III. 

1 

V3^ 


2 
sec  ^  = — ,     CSC  ^  =  ±  2. 

Vs 

(6)  30°,  90°,  150°,  270°. 
6.   161.41,  33°  34' 5'',  99°  4' 43''.  7.   69.812  yds. 

IV. 

6.   230.03  feet.        7.   ^  =  37°  24' 58",  B  =  51°  37' 52",  C  =  90°  57' 10' 


1.  17|  years.  4.    1. 

2.  siii2x=:±m,  tan2x=±-p^=-       ^'    1-7208. 

VI  -  m2         6^   j^^  50O 18'  E.,  399  mUes. 

3.  X  =  210°,  330°,  44°  25' 30",  135°  34' 30". 

VI. 

1.    16.                                                       4.  45°,  225°,  116°  33' 54", 

'    3tana:-tan3x                                .  296°  33' 54". 

1  —  3 tannic     '                              ^-  ^irst  ship,  223  miles;  second 

3.   Third  side,  any  value  ;  opposite  ^^^P'  ^^^  °^^®s- 

side,  13.766.                                  6.  0. 

VII. 

1.  25.  4.    ±90°,  180°. 

2.  2.  5.    S.  83°41'  E.;   1907  miles. 

3.  8.6816,    -5^,   43°  43' 10",    106°  16' 50". 


VIII. 


1.   27.  2.    a  =  V2Ftan^,  6  =  V2Fcot^. 

Z.    a-±  45°,  ±  135°;   &  =  ±  30°,  ±  150°. 

4.  Smallest  value  of  opposite  side,  1  ;  1.75,  53°  7'  48",  81°  52'  12"  or 

0.25,  126°  52'  12",  8°  7'  48". 

5.  39°  29'  N.,  67°  14'  W.  6.   tan  a  -  tan2  6  or  -  cot2&. 


18  TRIGONOMETRY. 

IX. 

1.  15.849. 

2.  a  =  2(3  +  V3),  6  =  2(V3  +  1),  c  =  4(V3  +  l),  B  =  60°. 

3.  155°  42'  20'^  114°  IT  42''. 

4.  41°  24'  35",  82°  49'  10",  55°  46'  15". 

6.   N.  69°56'E.;  609  miles.  6.    1. 

X. 

1.  1.23138.  4.    5,743^  4  357. 

2.  a  =  4,  6  =  3,  c  =  5,  ^  =  53°  7'  48",       5.    14°  10'  E. ;  342  miles. 

B  =  36°  52'  12".  6.    2. 

3.  cos2^. 

XI. 

1.   logs  4  =  1-.  4.    114.92  feet. 

3.    0.039345,  0.055226,  97°  45'.  5.    47°24'N.;   63°  43' W. 

XII. 

7t                         V3  — 1 
^'    3  ^-    — 2 ^-   462.34,  61°  37' 30",  56°  14' 30' 


XIII. 

12- 


1-   ^-  7.    188,280.  8.   44°  35' 40". 


XIV. 

1.   200°  32' 7".  5.    1, 

7.  a  =  273.76,  6  =  272.94,  c  =  256.65, 
a  =  62°  9' 42",     ^  =  61°  50' 18",       7  =  56°. 

8.  42°  49' 48". 

XV. 

1.  (a)  114°  35' 30",    (6)  |.  6.    205°  24' 47". 

7.    461.94;   59°  11' 8". 


ANSWERS. 


19 


Exercise  XXIII. 


1. 

Iogio6 

=  0.77815. 

logio  14 

=  1.14613. 

logio  21 

=  1.32222. 

logio4 

=  0.60206. 

logio  12 

=  1.07918. 

logio  5 

=  0.69897. 

logioi 

=  T.69897. 

logio  i 

=  1.39794. 

logio  1 

=  1.89086. 

logio  U 

=  0.02119. 

2. 

logs  10 

=  3.3219. 

log2  5 

=  2.3219. 

logs  5 

=  1.4650. 

10g7i 

=  -  0.3562. 

10g5  3l3 

=  -  2.2620. 

3. 

l0ge2 

=  0.69315. 

loge  3 

=  1.09861. 

loge  5 

=  1.60944. 

10ge7 

=  1.94591. 

loge  8 

=  2.07944. 

loge  9 

=  2.19722. 

logel- 

=  -  0.40546. 

logef 

=  -  0.22314. 

loge  If 

=  0.25952. 

logeeV 

=  -2.14843. 

4. 

x=  1.54396.              X 

=  0.83048. 

X  =  0.42062. 

Exercise  XXIV. 

1. 

lOgeS 

=  1.09861. 

loge  5 

=  1.60944. 

loge  7 

=  1.94591. 

2. 

loge  10 

=  2.3025850930. 

3. 

logio2 

=  0.30103. 

logioe 

=  0.43429. 

logio  11 

=  1.04139. 

1.  sin  V  =  0.00029088820. 
tan  1'  =  0.000290888212. 

2.  sin  2'  =  0.000581776. 


Exercise  XXV. 

cos  r  =  0.99999995769. 


3.    sin  1°  =  0.0175. 


6.   0°40'9" 


Exercise  XXVI. 


1.  sin  6' =  0.0017453 

2.  sin  2°  =  0.034902  ; 
sin  3°  =  0.052340 ; 
sin  4°  =0.069762; 


cos  G'  =  0.9999995. 
cos  2°  =  0.999392, 
cos  3°  =0.998632. 
cos  4°  =  0.997568. 


Exercise  XXVII. 


1.   The  6  sixth  roots  of  —  1  are  : 

V3  +  i       .      -\/3  +  i      -V3 


2  '  2 

The  6  sixth  roots  of  +  1  are : 

,      1  +  V^      -  1  +  V^ 
1.      7, '     7, 


-1, 


i, 

V3- 

-  i 

2 

_ 

1- 

-V- 

"3 

v:^ 


20  SPHERICAL    TRIGONOMETRY. 

2.  -^— '  -^ ,        -I. 

3.  cos  67i°  +  i  sin  67i°,  cos  157i°  +  i  sin  157^°,  cos  24 7^°  +  i  sin  247^'^ 
cos  337i°  +  i  sin  337i°. 

4.  sin  4  ^  =  4  cos^  ^  sin  ^  —  4  cos  6  sin^  ^. 
cos  4  ^  =  cos*  d  —  Q  cos2  ^  sin2  d  +  sin*  ^. 

Exercise  XXVIII. 

6.   secx  =  l  +  ^%^  +  ^V 

2  ^  24  ^  720 

6.  xcotx=l-f-^-|^^- 

3      45      945 

7.  sin  10°  =  0.173648,     cos  10°  =  0. 984808. 

8.  tanl5°  =  0.267958. 


SPHERICAL   TRIGONOMETRY. 

Exercise  XXIX. 

1.   110°,  100°,  80°        2.    140°,  90°,  55°.  7.    |  7t  ft.,  2  ;r  ft.,  2/;r  ft. 

Exercise  XXX. 

3,    (i.)  Either  a  or  6  must  be  equal  to  90°.  (lii.)  A  =  90°,  B  =  h. 

(ii.)  A  =  90°  and  B  =  b.  (iv.)  c  =  90°,  b  =  B  =  90° 

Exercise  XXXI. 

2.   I.  The  cosine  of  the  middle  part  =  the  product  of  the  cotangents  of 
the  adjacent  parts. 
II.  The  cosine  of  the  middle  part  =  the  product  of  the  sines  of  oppo- 
site parts. 


ANSWERS.  21 


Exercise  XXXII. 

24.  A  -  175°  57'  10'',  B  =  135°  42'  50",  C  =  135°  34'  7". 

25.  C=  104°  41' 39",    a  =  104°  53' 2",      6  =  133°  39' 48". 

26.  a  =  90° ;  b  and  B  are  indeterminate. 

27.  a  =  ^  =  60°,  6  =  90°,  B  =  90. 

28.  The  triangle  is  impossible. 

29.  b  =  130°  41'  42",  c  =  71°  27'  43",  A  =  112°  57'  2". 

30.  a  =  26°  3'  61",  A  =  35°,  B  =  65°  46'  7". 

31.  Impossible. 

Exercise  XXXIII. 

1,  cos  ^  =  cot  a  tan  ^6,  sin  1 5  =  esc  a  sin  I  6,  cos  ^  =  cos  a  sec  |  &. 

2,  sin  i  J.  =  ^  sec  ^  a. 

o      •     1    .  1  180°     .     ^       .     1  180° 

3,  sm  I  A  =  sec  1  a  cos ,  sm  R  =  sm  i  a  esc  — » 

sm  r  =  tan  I  a  cot 

^  n 

4,  Tetrahedron,    70°  31'  46";    octahedron,    109°  28'  14";    icosahedron, 

138°  11'  36";  cube,  90°;  dodecahedron,  116°  33'  44". 

5,  cot  I  A  =  Vcos  a. 


Exercise  XXXV. 

1.  (i.)  tan  m  =  tan  6  cos^,       (ii.)  tan  m  =  tan  c  cos  B, 

cos  a  =  cos  b sec  m  cos  {c  —  m);       cos b  =  cos  c  sec  »w  cos  (a  —  m). 


Exercise  XXXVI. 

1.    (i.)  cot  X  =  tan  B  esc  a,  (ii.)  cot  a;  =  tan C  esc  b, 

cos ^  =  cos B CSC X sin  (O  —  x) ;      cos5  =  cos  C  esc x  sin  (4  —  x). 


Exercise  XLI. 
4.   2066.5  square  miles. 


22  SPHERICAL    TRIGONOMETRY. 


Exercise  XLII. 

1.  If  X  denotes  the  angle  required,  sin  |x  =  cos  18°  sec  9°,  x  —  148°  42'. 

2.  cos  x—Q.0^  A  cos  B. 

3.  Let  w  =  the  inclination  of  the  edge  c  to  the  plane  of  a  and  b.     Then 

it  is  easily  shown  that  V=  abc  sin  I  sin  w.  Now,  conceive  a  sphere 
constructed  having  for  centre  the  vertex  of  the  trihedral  angle 
whose  edges  are  a,  6,  c.  The  spherical  triangle,  whose  vertices 
are  the  points  where  a,  b,  c  meet  the  surface  of  this  sphere,  has 
for  its  sides  I,  m,  n;  and  w  is  equal  to  the  perpendicular  arc  from 
the  side  I  to  the  opposite  vertex.  Let  L,  If,  N  denote  the  angles 
of  this  triangle.     Then,  by  means  of  [39]  and  [48],  we  find  that 

sin  w  =  smmsin  N=2 sin m sin  i  JV cos  i N 

2       / 

=  -: — :  Vsin  s  sin  is  —  I)  sin  (s  —  m)  sin  (s  —  n), 

where         s=-  \{l  +  in-\-  n)  -, 


hence,       ¥=2  abc  Vsin  s  sin  (s  —  I)  sin  (s  —  m)  sin  (s  —  n). 
4.    (i.)  9,976,500  square  miles ;    (ii.)  13,316,560  square  miles.  • 
6.   Let  m  =  longitude  of  point  where  the  ship  crosses  the  equator,  B  = 

her  course  at  the  equator,  d  =  distance  sailed.     Then 

tan  m  =  sin  I  tan  a,  cos  B  =  cos  I  sin  a,  cot  d  =  cot  I  cos  a. 

6.  Let  k  =  arc  of  the  parallel  between  the  places,  x  =  difference  required ; 

then  sin  |  fc  =  sin  i  d  sec  I.     x  —  90°(  V2  —  1). 

7.  tan  \{m  —  mf)  —  Vsec  s  sec  (s  —  d)  sin  (s  —  Z)  sin  (s  —  T) ;  where  2  s  = 

Z  +  r  +  d,  and  7m  and  m'  are  the  longitudes  of  the  places. 
9.   44  min.  past  12  o'clock.  10.   60°. 

11.  cos  i  =  —  tan  d  tan  I ;  time  of  sunrise  =  12  —  --  o'clock  a.m.  ;  time 

t  ^^ 

of  sunset  =  —z  o'clock  p.m.  ;  cos  a  =  sin  d  sec  I.     For  longest  day 
15 

at  Boston :  time  of  sunrise,  4  hrs.  26  min.  50  sec.  a.m.  ;  time  of 

sunset,  7  hrs.  33  min.  10  sec.  p.m.     Azimuth  of  sun  at  these  times, 

57°  25'  15'' ;  length  of  day,  15  hrs.  6  min.  20  sec.  ;  for  shortest  day, 

times  of  sunrise  and  sunset  are  7  hrs.  33  min.  10  sec.  a.m.  and 

4  hrs.  26  min.  50  sec.  p.m.  ;  azimuth  of  sun,  122°  34' 45";  length 

of  day,  8  hrs.  53  min.  40  sec. 

12.  The  problem  is  impossible  when  cot  d  <!  tan  I ;  that  is,  for  places  in 

the  frigid  zone. 


ANSWERS.  23 

13.  For  the  northern  hemisphere  and  positive  declination, 

sin  h  =  sinl  sin  d,  cot  a  =  cos  I  tan  d. 
Example  :  ^  =  17°  14'  35'^  a  =  73°  51'  34''  E. 

14.  The  farther  the  place  from  the  equator,  the  greater  the  sun's  altitude 

at  6  A.M.  in  summer.  At  the  equator  it  is  0°.  At  the  north  pole 
it  is  equal  to  the  sun's  declination.  At  a  given  place,  the  sun's 
altitude  at  6  a.m.  is  a  maximum  on  the  longest  day  of  the  year, 
and  then  sin  h  =  sinl  sin  e  (where  e  =  23°  27'). 

15.  cos  t  =  coil  tan  d.      Times  of  bearing  due  east  and  due  west  are 

12  —  —  o'clock  A.M.,  and  —  o'clock  p.m.,  respectively, 
io  lu 

Example  :  6  hrs.  58  min.  a.m.  and  5  hrs.  2  min.  p.m. 

16.  When  the  days  and  nights  are  equal,  d  =  0°,  cos  t  =  0,  t  =  90°  ;  that 

is,  sun  is  everywhere  due  east  at  6  a.m.,  and  due  west  at  6  p.m. 
Since  I  and  d  must  both  be  less  than  90°,  cos  t  cannot  be  negative, 
therefore  t  cannot  be  greater  than  90°.  As  d  increases,  t  decreases  ; 
that  is,  the  times  in  question  both  approach  noon.  If  Z  <  d,  then 
cos  it>l ;  therefore  this  case  is  impossible.  If  Z  =  d,  then  cos  <  =  1, 
and  t  =  0°  ;  that  is,  the  times  both  coincide  with  noon.  The  ex- 
planation of  this  result  is,  that  for  d  =  I  the  sun  at  noon  is  in  the 
zenith,  and  south  of  the  prime  vertical  at  every  other  time.  And 
if  Z  >  d,  the  diurnal  circle  of  the  sun  and  the  prime  vertical  of  the 
place  meet  in  two  points  which  separate  further  and  further  as  I 
increases.  At  the  pole  the  prime  vertical  is  indeterminate ;  but 
near  the  pole,  t  =  90°,  and  the  sun  is  always  east  at  6  a.m. 

17.  sin  Z  =:  sin  d  esc /i.  18.    11°  50' 35". 

19.  The  bearing  of  the  wall,  reckoned  from  the  north  point  of  the  hori- 

zon, is  given  by  the  equation  cot  x  =  cosl  tan  d  ;  whence,  for  the 
given  case,  x  =  75°  12'  38". 

20.  55°  45'  6"  N.  21.    63°  23'  41"  N.  or  S. 

22.  (i.)  cost  —  —  tan  I  coip  ;  (ii.)  t  =  z  ;  (iii.)  the  result  is  indeterminate. 

23.  cot  a  =  cos  Z  tan  (i.  28.    sin  d  =  sin  e  sin  w,  tan  r  =  cos  e  tan  w. 

25.  h  =  65°  37'  20".  29.    d  =  32°  24'  12",  r  =  301°  48'  17". 

26.  h  =  58°  25'  15",  a  =  152°  28'.    30.    d  =  20°  48'  12". 

27.  t  =  45°  42',  I  =  67°  58'  54".     31.   3  hrs.  59  min.  27|  sec.  p.m. 


32.   cos  I  a  =  Vcos  |  (l  +  h  +  p)cosl{l  +  h  — p)  sec  I  sec  h. 


FIVE -PLACE 


LOGAEITHMIC  AND  TRIGONOMETRIC 


TABLES 


ARRANGED    BY 


G.   A.   WENTWORTH,   A.M. 


G.   A.   HILL,   A.M. 


Boston,  U.S.A.,  and  London 

PUBLISHED  BY   GINN   &   COMPANY 

1897 


Entered  according  to  Act  of  Congress,  in  the  year  1882,  by 

G.  A.  WENTWORTH  and  G.  A.  HILL 
in  the  office  of  the  Librarian  of  Congress  at  Washington 


Copyright,  1895,  by  G.  A.  Wkntworth  and  G.  A.  Hill. 


INTRODUCTION. 


1.  If  the  natural  numbers  are  regarded  as  powers  of  ten,  tlie 
exponents  of  the  powers  are  the  Common  or  Briggs  Logarithms  of 
the  numbers.  If  A  and  B  denote  natural  numbers,  a  and  b  their 
logarithms,  then  10"  =  A,  10*  =  -5 ;  or,  written  in  logarithmic  form, 

log^  =  a,         logB^b.  .  ' 

2.  The  logarithm  of  a  product  is  found  by  adding  the  logarithms 
of  its  factors. 

For,  Ax  B  =  10«  X  10»  =  10«  +  ». 

Therefore,  log{A  x  B)  =  a-\-b  =  \ogA-{-  log  B. 

3.  The  logarithm  of  a  quotient  is  found  by  subtracting  the 
logarithm  of  the  divisor  from  that  of  the  dividend. 

Therefore,  log  —  =  a  —  6  =  log  J.  —  log  B. 

n 

4.  The  logarithm  of  a  power  of  a  number  is  found  by  multiply- 
ing the  logarithm  of  the  number  by  the  exponent  of  the  power. 

For,  ^«  =  (10«)'»  =  10«». 

Therefore,  log  J."  —  an  =  n  log  A. 

5.  The  logarithm  of  the  root  of  a  number  is  found  by  dividing 
the  logarithm  of  the  number  by  the  index  of  the  root. 

For,  \C4  =  -7To«  =  10*. 

Therefore,  log  \(I  =  -  =  -^^—  • 


6.  The  logarithms  of  1,  10,  100,  etc.,  and  of  0.1,  0.01,  0.001, 
etc.,  are  integral  numbers.  The  logarithms  of  all  other  numbers 
are  fractions. 


IV  LOGARITHMS. 


For,    100  =       1,  hence       log  1  =  0 

101  =      10,  hence      log  10  =  1 

102  =    100,  hence    log  100  =  2 

103  =  1000,  hence  log  1000  =  3 


10-1  =      0. 1,  hence      log  0. 1  =  —  1 ; 

10-2  =    0.01,  hence    log  0.01  =  —  2  ; 

10-^  =  0.001,  hence  log  0.001  =  —  3  ; 
and  so  on. 

If  the  number  is  between  1  and  10,  the  logarithm  is  between  0  and  1. 
If  the  number  is  between  10  and  100,  the  logarithm  is  between  1  and  2. 
If  the  number  is  between  100  and  1000,  the  logarithm  is  between  2  and  3. 
If  the  number  is  between  1  and  0.1,  the  logarithm  is  between  0  and  —1. 
If  the  number  is  between  0.1  and  0.01,  the  logarithm  is  between  —1  and  —2. 
If  the  number  is  between  0.01  and  0.001,  the  logarithm  is  between  —2  and  —3, 
And  so  on. 

7.  If  the  number  is  less  than  1,  the  logarithm  is  negative  (§  6), 
but  is  written  in  such  a  form  that  t\iQ  fractional  part  is  dXwdt>j^  positive. 

For  the  number  may  be  regarded  as  the  product  of  two  factors,  one  of 
which  lies  between  1  and  10,  and  the  other  is  a  negative  power  of  10 ;  the 
logarithm  will  then  take  the  form  of  a  difference  whose  minuend  is  a  positive 
proper  fraction,  and  whose  subtrahend  is  a  positive  integral  number. 

Thus,  0.48  =  4.8X0.1. 

Therefore  (§  2),  log     0.48  =  log  4.8  +  log  0.1  =  0.68124  -  1.     (Page  1.) 

Again,  0.0007  =  7  X  0.0001. 

Therefore,  log  0.0007  =  log  7  +  log  0.0001  =  0.84510  —  4. 

The  logarithm  0.84510  —  4  is  often  written  4.84510. 

8.  Every  logarithm,  therefore,  consists  of  two  parts  :  a  positive 
or  negative  integral  number,  which  is  called  the  Characteristic,  and 
a  positive  proper  fraction,  which  is  called  the  Mantissa. 

Thus,  in  the  logarithm  3.52179,  the  integral  number  3  is  the  characteristic, 
and  the  fraction  .52179  the  mantissa.  In  the  logarithm  0.78254  —  2,  the  inte- 
gral number  —  2  is  the  characteristic,  and  the  fraction  0.78254  is  the  mantissa. 

9.  If  the  logarithm  is  negative,  it  is  customary  to  change  the 
form  of  the  difference  so  that  the  subtrahend  shall  be  10  or  a  multiple 
of  10.  This  is  done  by  adding  to  both  minuend  and  subtrahend  a 
number  which  will  increase  the  subtrahend  to  10  or  a  multiple  of  10. 

Thus,  the  logarithm  0.78254  —  2  is  changed  to  8.78254  —  10  by  adding  8  to 
both  minuend  and  subtrahend.  The  logarithm  0.92737  —  13  is  changed  to 
7.92737  —  20  by  adding  7  to  both  minuend  and  subtrahend. 

10.  The  following  rules  are  derived  from  §  6  :  — 

If  the  number  is  greater  than  1,  make  the  characteristic  of  the 
logarithm  one  unit  less  than  the  number  of  figures  on  the  left  of 
the  decimal  point. 

If  the  number  is  less  than  1,  make  the  characteristic  of  the  loga- 
rithm negative,  and  one  unit  vnore  than  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  given  number. 


INTRODUCTION.  V 

If  the  characteristic  of  a  given  logarithm  is  positive,  make  the 
number  of  figures  in  the  integral  part  of  the  corresponding  number 
one  more  than  the  number  of  units  in  the  characteristic. 

If  the  characteristic  is  negative,  make  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  correspond- 
ing number  one  less  than  the  number  of  units  in  the  characteristic. 

Thus,  the  characteristic  of  log  7849.27  =  3  ; 

the  characteristic  of  log  0.037  =  —  2  =  8.00000  —  10. 
If  the  characteristic  is  4,  the  corresponding  number  has  five  figures  in  its  inte- 
gral part.     If  the  characteristic  is  —  3,  that  is,  7.00000  —  10,  the  corresponding 
fraction  has  two  zeros  between  the  decimal  point  and  the  first  significant  figure. 

11.  The  logarithms  of  numbers  that  can  be  derived  one  from 
another  by  multiplication  or  division  by  an  integral  power  of  10 
have  the  same  mantissa. 

For,  multiplying  or  dividing  a  number  by  an  integral  power  of  10  will 
increase  or  diminish  its  logarithm  by  the  exponent  of  that  power  of  10 ;  and 
since  this  exponent  is  an  integer,  the  mantissa  of  the  logarithm  will  be 
unaffected. 

Thus,        log  4.6021      =0.66296.     (Page  9.) 

log  460.21      =  log  (4.6021  X  102)  =  log  4.6021  +  log  10^ 

=  0.66296  +  2  =  2.66296. 
log  460210     =  log  (4.6021  X  10^)  =  log  4.6021  +  log  10^ 

=  0.66296  +  5  =  6.66296. 
log  0.046021  =  log  (4.6021  -^  102)  =  log  4.6021  -  log  102 

=  0.66296  -  2  =  8.66296  -  10. 


TABLE  I. 

12.  In  this  table  (pp.  1-19)  the  vertical  columns  headed  N  con- 
tain the  numbers,  and  the  other  columns  the  logarithms.  On  page  1 
both  the  characteristio  and  the  mantissa  are  printed.  On  pages 
2-19  the  mantissa  only  is  printed. 

The  fractional  part  of  a  logarithm  can  be  expressed  only  approx- 
imately, and  in  a  five-place  table  all  figures  that  follow  the  fifth  are 
rejected.  Whenever  the  sixth  figure  is  5,  or  more,  the  fifth  figure 
is  increased  by  1.  The  figure  5  is  written  when  the  value  of  the 
figure  in  the  place  in  which  it  stands,  together  with  the  succeeding 
figures,  is  more  than  4J,  but  less  than  5. 

Thus,  if  the  mantissa  of  a  logarithm  written  to  seven  places  is  5328732,  it  is 
written  in  this  table  (a  five-place  table)  53287.  If  it  is  6328751,  it  is  written 
63288.     If  it  is  6328461  or  5328499,  it  is  written  in  this  table  53285. 

Again,  if  the  mantissa  is  5324981,  it  is  written  63260 ;  and  if  it  is  4999967,  it 
is  written  60000. 


VI  LOGARITHMS. 

This  distinction  between  5  and  5,  in  case  it  is  desired  to  curtail 
still  further  the  mantissas  of  logarithms,  removes  all  doubt  whether 
a  5  in  the  last  given  place,  or  in  the  last  but  one  followed  by  a 
zero,  should  be  simply  rejected,  or  whether  the  rejection  should 
lead  us  to  increase  the  preceding  figure  by  one  unit. 

Thus,  the  mantissa  1392^  when  reduced  to  four  places  should  be  1392  ;  but 
13925  should  be  1393. 

To  Find  the  Logarithm  of  a  Given  Number. 

13.  If  the  given  number  consists  of  one  or  two  significant 
figures,  the  logarithm  is  given  on  page  1.  If  zeros  follow  the 
significant  figures,  or  if  the  number  is  a  proper  decimal  fraction, 
the  characteristic  must  be  determined  by  §  10. 

14.  If  the  given  number  has  three  significant  figures,  it  will  be 
found  in  the  column  headed  N  (pp.  2-19),  and  the  mantissa  of  its 
logarithm  in  the  next  column  to  the  right,  and  on  the  same  line. 
Thus, 

Page    2.     log  145  =  2.16137,  log  14500  =  4.16137. 

Page  14.    log  716  =  2.85491,  log  0.716  =  9.85491  -  10. 

15.  If  the  given  number  has  four  significant  figures,  the  first 
three  will  be  found  in  the  column  headed  N,  and  the  fourth  at  the 
top  of  the  page  in  the  line  containing  the  figures  1,  2,  3,  etc.  .  The 
mantissa  will  be  found  in  the  column  headed  by  the  fourth  figure, 
and  on  the  same  line  with  the  first  three  figures.     Thus, 

Page  15.     log  7682    =  3.88547,        log  76.85    =  1.88564. 
Page  18.     log  93280  =  4.96979,         log  0.9468  =  9.97626  —  10. 

16.  If  the  given  number  has  five  or  more  significant  figures,  a 
process  called  interpolation  is  required. 

Interpolation  is  based  on  the  assumption  that  between  two  con- 
secutive mantissas  of  the  table  the  change  in  the  mantissa  is  directly 
proportional  to  the  change  in  the  number. 

Kequired  the  logarithm  of  34237. 

The  required  mantissa  is  (§  11)  the  same  as  the  mantissa  for  3423.7  ;  there- 
fore it  will  be  found  by  adding  to  the  mantissa  of  3423  seven-tenths  of  the 
difference  between  the  mantissas  for  3423  and  3424. 

The  mantissa  for  3423  is  53441. 

The  difference  between  the  mantissas  for  3423  and  .3424  is  12. 

Hence,  the  mantissa  for  3423.7  is  53441  -f-  (0.7  X  12)  =  53449. 

Therefore,  the  required  logarithm  of  34237  is  4.53449. 


C^^ 


INTRODUCTION.  Vll 

Eequired  the  logarithm  of  0.0015764. 

The  required  mantissa  is  the  same  as  the  mantissa  for  1576.4 ;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  1576  four-tenths  of  the  difference 
between  the  mantissas  for  1576  and  1577. 

The  mantissa  for  1576  is  19756. 

The  difference  between  the  mantissas  for  1576  and  1577  is  27. 

Hence,  the  mantissa  for  1576.4  is  19756  +  (0.4  X  27)  =  19767. 

Therefore,  the  required  logarithm  of  0.0015764  is  7.19767  —  10. 

Eequired  the  logarithm  of  32.6708. 

The  required  mantissa  is  the  same  as  the  mantissa  for  3267.08 ;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  3267  eight-hundredths  of  the 
difference  between  the  mantissas  for  3267  and  3268. 

The  mantissa  for  3267  is  51415. 

The  difference  between  the  mantissas  for  3267  and  3268  is  13. 

Hence,  the  mantissa  for  3267.08  is  51415  +  (0.08  X  13)  =  51416. 

Therefore,  the  required  logarithm  of  32.6708  is  1.51416. 

17.  When  the  fraction  of  a  unit  in  the  part  to  be  added  to  the 
mantissa  for  four  figures  is  less  than  0.5  it  is  to  be  neglected ;  when 
it  is  0.5  or  more  than  0.5  it  is  to  be  taken  as  one  unit. 

Thus,  in  the  first  example,  the  part  to  be  added  to  the  mantissa  for  3423  is 
8.4,  and  the  .4  is  rejected.  In  the  second  example,  the  part  to  be  added  to  the 
mantissa  for  1576  is  10.8,  and  11  is  added. 

To  Find   the  Antilogarithm  ;    that  is,  the  Number  Corre- 
sponding TO  A  Given  Logarithm. 

18.  If  the  given  mantissa  can  be  found  in  the  table,  the  first 
three  figures  of  the  required  number  will  be  found  in  the  same  line 
with  the  mantissa  in  the  column  headed  N,  and  the  fourth  figure  at 
the  top  of  the  column  containing  the  mantissa. 

The  position  of  the  decimal  point  is  determined  by  the  charac- 
teristic (§  10). 

Find  the  number  corresponding  to  the  logarithm  0.92002. 

Page  16.     The  number  for  the  mantissa  92002  is  8318. 

The  characteristic  is  0 ;   therefore,  the  required  number  is  8.318. 

Find  the  number  corresponding  to  the  logarithm  6.09167. 

Page  2.     The  number  for  the  mantissa  09167  is  1235. 

The  characteristic  is  6 ;  therefore,  the  required  number  is  1235000. 

Find  the  number  corresponding  to  the  logarithm  7.50325  — 10. 

Page  6.     The  number  for  the  mantissa  50325  is  3186. 

The  characteristic  is  —  3  ;  therefore,  the  required  number  is  0.003186, 


Vlll  LOGARITHMS. 

19.  If  the  given  mantissa  cannot  be  found  in  the  table,  find 
in  the  table  the  two  adjacent  mantissas  between  which  the  given 
mantissa  lies,  and  the  four  figures  corresponding  to  the  smaller  of 
these  two  mantissas  will  be  the  first  four  significant  figures  of  the 
required  number.  If  more  than  four  figures  are  desired,  they  may- 
be found  by  interpolation,  as  in  the  following  examples  : 

Find  the  number  corresponding  to  the  logarithm  1.48762. 

Here  the  two  adjacent  mantissas  of  the  table,  between  which  the  given  man- 
tissa 48762  lies,  are  found  to  be  (page  6)  48756  and  48770.  The  corresponding 
numbers  are  3073  and  3074.  The  smaller  of  these,  3073,  contains  the  first  four 
significant  figures  of  the  required  number. 

The  difference  between  the  two  adjacent  mantissas  is  14,  and  the  difference 
between  the  corresponding  numbers  is  1 . 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas,  48756, 
and  the  given  mantissa,  48762,  is  6.  Therefore,  the  number  to  be  annexed  to 
3073  is  j'^j  of  1  =  0.428,  and  the  fifth  significant  figure  of  the  required  number 
is  4. 

Hence,  the  required  number  is  30.734. 

Find  the  number  corresponding  to  the  logarithm  7.82326  — 10. 

The  two  adjacent  mantissas  between  which  82326  lies  are  (page  13)  82321 
and  82328.     The  number  corresponding  to  the  mantissa  82321  is  6656. 

The  difference  between  the  two  adjacent  mantissas  is  7,  and  the  difference 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  mantissa,  82321,  and  the  given  mantissa, 
82326,  is  5.  Therefore,  the  number  to  be  annexed  to  6666  is  f  of  1  =  0.7,  and 
the  fifth  significant  figure  of  the  required  number  is  7. 

Hence,  the  required  number  is  0.0066567. 

In  using  a  five-place  table  the  numbers  corresponding  to  man- 
tissas may  be  carried  to  five  significant  figures,  and  in  the  first 
part  of  the  table  to  six  figures.* 

20.  The  logarithm  of  the  reciprocal  of  a  number  is  called  the 
Cologarithm  of  the  number. 

If  A  denotes  any  number,  then 

colog  ^  =  log  —  =  log  1  —  log  ^  (§  3)  =  —  log  A. 

Hence,  the  cologarithm  of  a  number  is  equal  to  the  logarithm  of 
the  number  with  the  minus  sign  prefixed,  which  sign  affects  the 
entire  logarithm,  both  characteristic  and  mantissa. 

*In  most  tables  of  logarithms  proportional  parts  are  given  as  an  aid  to 
interpolation  ;  but,  after  a  little  practice,  the  operation  can  be  performed  nearly 
as  rapidly  without  them.  Their  omission  allows  a  page  with  larger-faced  type 
and  more  open  spacing,  and  consequently  less  trying  to  the  eyes. 


INTRODUCTION.  IX 

In  order  to  avoid  a  negative  mantissa  in  the  cologarifhin,  it  is 
customary  to  substitute  for  —  log  A  its  equivalent 

(10  -  log  ^)  — 10. 

Hence,  the  cologarithm  of  a  number  is  found  by  subtracting  the 
logarithm  of  the  number  from  10,  and  then  annexing  — 10  to  the 
remainder. 

The  best  way  to  perform  the  subtraction  is  to  begin  on  the  left 
and  subtract  each  figure  of  log  A  from  9  until  we  reach  the  last 
significant  figure,  which  must  be  subtracted  from  10. 

If  log  A  is  greater  in  absolute  value  than  10  and  less  than  20, 
then  in  order  to  avoid  a  negative  mantissa,  it  is  necessary  to  write 
—  log  A  in  the  form 

(20  -  log  ^)- 20. 

So  that,  in  this  case,  colog  A  is  found  by  subtracting  log  A  from 
20,  and  then  annexing  —  20  to  the  remainder. 

Find  the  cologarithm  of  4007. 

10  -10 

Page  8.  log  4007  =    3.60282 

colog  4007=    6.39718  —  10 

Find  the  cologarithm  of  103992000000. 

20  -20 

Page  2.     log  103992000000  =  11.01700 

colog  103992000000  =    8.98300  -  20 

If  the  characteristic  of  log  A  is  negative,  then  the  subtrahend, 
— 10  or  —  20,  will  vanish  in  finding  the  value  of  colog  A. 

Find  the  cologarithm  of  0.004007. 

10  -10 

log  0.004007  =    7.60282  -  10 
colog  0.004007  =    2.39718 

With  practice,  the  cologarithm  of  a  number  can  be  taken  from 
the  table  as  rapidly  as  the  logarithm  itself. 

By  using  cologarithms  the  inconvenience  of  subtracting  the  log- 
arithm of  a  divisor  is  avoided.  For  dividing  by  a  number  is 
equivalent  to  multiplying  by  its  reciprocal.  Hence,  instead  of 
subtracting  the  logarithm  of  a  divisor  its  cologarithm  may  be 
added. 


LOGARITHMS. 


r 


Find  the  logarithms  of 


Exercises. 


1.  6170. 

2.  0.617. 

3.  2867. 


4.  85.76. 

5.  296.8. 

6.  7004. 


7.  0.8694. 

8.  0.5908. 

9.  73243. 


10.  67.3208. 

11.  18.5283. 

12.  0.0042003. 


Find  the  cologarithms  of  : 


13.  72433. 

14.  802.376. 

15.  15.7643. 


16.  869.278. 

17.  154000. 

18.  70.0426. 


19.  0.002403. 

20.  0.000777. 

21.  0.051828. 


Find  the  antilogarithms  of : 


22.  2.47246. 

23.  7.89081. 

24.  '2.91221. 


25.  1.26784. 

26.  3.79029. 

27.  6.18752. 


28.  9.79029-10. 

29.  7.62328-10. 

30.  6.16465-10. 


Computation  by  Logarithms. 

21.    (1)  Find  the  value  of  x,  ii  x  =  72214  X  0.08203. 

Page  14.  log  72214     =  4.85862 

Page  16.  log  0.08203  =  8.91397  -  10 

By  §2.  logx  =3.77259 

Page  11.  X  =  5923.63 


(2)  Find  the  value  of  x,  ii  x  =  5250  -^  23487. 

Page  10.  log  5250  =  3.72010 

Page  4.  colog  23487  =  5.62917  -  10 

Page  4.  log  X  =  9.34933  -  10  =  log  0.22353 

.-.  X  =  0.22353 


(3)  Find  the  value  of  x,  if  x^= 


7.56  X  4667  X  567 


Page  15. 
Page  9. 
Page  11. 
Page  17. 
Page  6. 
Page  4. 
Page  5. 


899.1  X  0.00337  X  23435 

log  7.56  =  0.87852 

log  4667  =  3.66904 

log  567  -  2.75358 

colog  899.1  =7.04619-10 
colog  0.00337  =  2.47237 

colog  23435  =  5.63013-10 

log  a;  =  2.44983  =  log  281.73 

.-. «  =281.73. 


INTRODUCTION.  XI 

(4)  Find  the  cube  of  376. 

Page  7.  log  376  =  2.57519 

Multiply  by  3  (§  4),  3 

Page  10.  log  3763  .  =  7.72557  =  log  53158600 

.-.  3763  =  53158600. 

(5)  Find  the  square  of  0.003278. 

Page  6.  log  0.003278  =    7.51561-10 

2 

Page  2.  log  0.0032782  =  15.03122  -  20  =  log  0.000010745 

.•.0.003278-2=    0.000010745. 

(6)  Find  the  square  root  of  8322. 

Page  16.  log  8322  =  3.92023 

Divide  by  2  (§  5),      2)3.92023 

log  V8322  =  1.96012  =  log  91.226 

.-.  V8322  =  91.226. 

If  the  given  number  is  a  proper  fraction,  its  logarithm  will  have 
as  a  subtrahend  10  or  a  multiple  of  10.  In  this  case,  before  divid- 
ing the  logarithm  by  the  index  of  the  root,  both  the  subtrahend  and 
the  number  preceding  the  mantissa  should  be  increased  by  such  a 
number  as  will  make  the  subtrahend,  when  divided  by  the  index  of 
the  root,  10  or  a  multiple  of  10. 

(7)  Find  the  square  root  of  0.000043641. 

Pages.  log  0.000043641     =    5.63989-10 

10  -10 

Divide  by  2  (§  5),      2)15.63989-20 

Page  13.  log  VO. 000043641  =    7.81995  -  10  =  log  0.0066062 

.-.  Vo.000043641  =  0.0066062. 

(8)  Find  the  sixth  root  of  0.076553. 

Page  15.  log  0.076553  =    8.88397  —  10 

50  -50 

Divide  by  6  (§  5),    ^  6)58.88397  -  60 

Page  13.  log  Vo. 076553       =    9.81400  -  10  =  log  0.65163 

.-.  \/0.076553       =    0.65163. 


Exercises. 


Find  by  logarithms  the  value  of : 

45607  5.6123  2.567 

31045'  0.01987*  0.05786 


XU  LOGARITHMS. 

0.06547 


4. 


5. 


7, 


74.938  X  0.05938 

4.657  X  0.03467 
3.908  X  0.07189'. 

0.0075389  X  0.0079 
0.00907  X  0  009784' 

312  X  7.18  X  31.82 
519  X  8.27  X  6.132' 


0.007X57.83X28.13      ^   ,00^   l  io  (>  ^ 

9.317  X  00.28  X  476.5  ' 

5.55  X  0.0007632  X  0.87654     ^        Ol  (^    i  ^    'T^^ 
2.79X0.0009524X1.46785*    ^     ^^  J      '    ' 

/0. 003457  X  43.387  X  99.2  X  0.00025       /~    ^  ^  /)  ^ 
\  0.005824  X  15.724  X  1.38  X  0.00089'    ^  •     /  ^      ^     f 


11       '/2' 


23.815  X  29.36  X  0.007  X  0.62487 
00072  X  9.236  X  5.924  X  3.0007 


1  .<)  i'  1  *• 


■-4 


/3.1416  X  0.031416  X  0.0031416 
'^'\7285  X  0.017285  X  0.0017285' 


TABLE  II. 


22.    This  table  (page  20)  contains  the  value  of  the  number  tt, 
its  most  useful  combinations,  and  their  logarithms. 

Find  the  length  of  an  arc  of  47°  32'  57'^  in  a  unit  circle. 


47°  32'  57''  =  171177" 

log  171177      =  5.23344 

log  \                     =  4.68557  - 

-10 

log  arc  47°  32'  57"  =  9.91901  -  10  =  log  0.82994 
.-.  length  of  arc       =  0.82994. 

Find  the  angle  if  the  length  of  its  arc  in  a  unit  circle  =  0.54936. 

log  0.54936  =9.73986-10 

colog  ^,  =  log  a"  =5.31443 

log  angle  =  5.05429  =  log  113316 

.-.  angle  =  113316"  =  31°  28'  36". 


INTRODUCTION.  XIU 

23.  The  relations  between  arcs  and  angles  given  in  Table  II. 
are  readily  deduced  from  the  circular  measure  of  an  angle. 

In  Circular  Measure  an  angle  is  defined  by  the  equation 

-  arc 

angle  =  —-r^ — , 
radius 

in  which  the  word  arc  denotes  the  length  of  the  arc  corresponding 
to  the  angle,  when  both  arc  and  radius  are  expressed  in  terms  of 
the  same  linear  unit. 

Since  the  arc  and  radius  for  a  given  angle  in  different  circles 
vary  in  the  same  ratio,  the  value  of  the  angle  given  by  this  equa- 
tion is  independent  of  the  value  of  the  radius. 

The  angle  which  is  measured  by  a  radius-arc  is  called  a  Radian, 
and  is  the  angular  unit  in  circular  measure. 

C  ^  C 

Since  C  =  2  irB,  it  follows  that  —  =  2  tt,  and  ~-  =  it.  Therefore, 

IC  K 

If  the  arc  =  circumference,  the  angle  =  2  tt. 

If  the  arc  ^  semicircumference,  the  angle  =  tt. 

If  the  arc  =  quadrant,  the  angle  =  ^  tt. 

If  the  arc  =  radius,  the  angle  =:  1. 

Therefore,  tt  =  180°,  \ir  =  90°,  i tt  =  60°,  i tt  =  45°,  i  tt  =  30°, 
■J  TT  =  22^°,  and  so  on. 

Since  180°  in  common  measure  equals  tt  units  in  circular  measure, 

77" 

1°  in  common  measure        =  t^^:  units  in  circular  measure  j 
1  unit  m  circular  measure  =  - —  m  common  measure. 

TT 

By  means  of  these  two  equations,  the  value  of  an  angle  expressed 

in  one  measure  may  be  changed  to  its  value  in  the  other  measure. 

Thus,  the  angle  whose  arc  is  equal  to  the  radius  is  an  angle  of 

180° 
1  unit  in  circular  measure,  and  is  equal  to  ,  or  57°  17'  45", 

TT 

very  nearly. 

TABLE  III. 

24.  This  table  (pp.  21-49)  contains  the  logarithms  of  the  trigo- 
nometric functions  of  angles.  In  order  to  avoid  negative  character- 
istics, the  characteristic  of  every  logarithm  is  printed  10  too  large. 
Therefore,  —10  is  to  be  annexed  to  each  logarithm. 

On  pages  28-49  the  characteristic  remains  the  same  throughout 
each  column,  and  is  printed  at  the  top  and  the  bottom  of  the  column. 


XIV  LOGARITHMS. 

But  on  pp.  30,  49,  the  characteristic  changes  one  unit  in  valtie  at  the 
places  marked  with  bars.  Above  these  bars  the  proper  characteristic 
is  printed  at  the  top,  and  below  them  at  the  bottom,  of  the  column. 

25.  On  pages  28-49  the  log  sin,  log  tan,  log  cot,  and  log  cos,  of 
1°  to  89°,  are  given  to  every  minute.  Conversely,  this  part  of  the 
table  gives  the  value  of  the  angle  to  the  nearest  minute  when 
log  sin,  log  tan,  log  cot,  or  log  cos  is  known,  provided  log  sin  or 
log  cos  lies  between  8.24186  and  9.99993,  and  log  tan  or  log  cot 
lies  between  8.24192  and  11.75808. 

If  the  exact  value  of  the  given  logarithm  of  a  function  is  not 
found  in  the  table,  the  value  nearest  to  it  is  to  be  taken,  unless 
interpolation  is  employed  as  explained  in  §  26. 

If  the  angle  is  less  than  45°  the  number  of  degrees  is  printed  at 
the  top  of  the  page,  and  the  number  of  minutes  in  the  column  to 
the  left  of  the  columns  containing  the  logarithm.  If  the  angle 
is  greater  than  45°,  the  number  of  degrees  is  printed  at  the  bottom 
of  the  page,  and  the  number  of  minutes  in  the  column  to  the  right 
of  the  columns  containing  the  logarithms. 

If  the  angle  is  less  than  45°,  the  names  of  its  functions  are 
printed  at  the  top  of  the  page ;  if  greater  than  45°,  at  the  bottom 
of  the  page.     Thus, 

Page  38.     log  sin  21°  37' =    9.56631  —  10. 

Page  45.     log  cot  36°  53'  =  10. 12473  -10  =  0. 12473. 

Page  37.     log  cos  69°  14'  =    9.54969  —  10. 

Page  49.     log  tan  45°  59'  =  10.01491  —  10  =  0.01491. 

Page  48.     If  log  cos  =  9.87468  —  10,  angle  =  41°  28'. 

Page  34.     If  log  cot  =  9.39353  -  10,  angle  =  76°  6'. 

If  log  sin  =  9.47760  —  10,  the  nearest  log  sin  in  the  table  is  9.47774  —  10 
(page  36),  and  the  angle  corresponding  to  this  value  is  17°  29'. 

If  log  tan  =  0.76520  =  10.76520  —  10,  the  nearest  log  tan  in  the  table  is 
10.76490  —  10  (page  32),  and  the  angle  corresponding  to  this  value  is  80°  15'. 

26.  If  it  is  desired  to  obtain  the  logarithms  of  the  functions  of 
angles  that  contain  seconds,  or  to  obtain  the  value  of  the  angle  in 
degrees,  minutes,  and  seconds,  from  the  logarithms  of  its  functions, 
interpolation  must  be  employed.  Here  it  must  be  remembered 
that. 

The  difference  between  two  consecutive  angles  in  the  table 
is  60". 

Log  sin  and  log  tan  increase  as  the  angle  increases ;  log  cos  and 
log  cot  diminish  as  the  angle  increases.. 


INTRODUCTIOK.  XV 

Find  log  tan  70°  46'  8". 

Page  37.    log  tan  70°  46'  =  0.45731. 

The  difference  between  the  mantissas  of  log  tan  70°  46'  and  log  tan  70°  47' 
is  41,  and  -^\  of  41  =  5. 

As  the  function  is  increasing,  the  5  must  be  added  to  the  figure  in  the  fifth 
place  of  the  mantissa  45731 ;   and 

Therefore  log  tan  70°  46'  8"  =  0.45736. 

Find  log  cos  47°  35'  4". 

Page  48.     log  cos  47°  35'  =  9.82899  -  10. 

The  difference  between  this  mantissa  and  the  mantissas  of  the  next  log  cos 
is  14,  and  q%  of  14  =  1. 

As  the  function  is  decreasing,  the  1  must  be  subtracted  from  the  figure  in  the 
fifth  place  of  the  mantissa  82899  ;   and 

Therefore  log  cos  47°  35'  4"  =  9.82898  -  10. 

Find  the  angle  for  which  log  sin  =  9.45359  — 10. 

Page  35.     The  mantissa  of  the  nearest  smaller  log  sin  in  the  table  is  45334. 

The  angle  corresponding  to  this  value  is  16°  30'. 

The  difference  between  45334  and  the  given  mantissa,  55359,  is  25. 

The  difference  between  45334  and  the  next  following  mantissa,  45377,  is  43, 
and  If  of  60"  =  35". 

As  the  function  is  increasing,  the  35"  must  be  added  to  16°  30';  and  the 
required  angle  is  16°  30'  35". 

Find  the  angle  for  which  log  cot  =  0.73478. 

Page  32.     The  mantissa  of  the  nearest  smaller  log  cot  in  the  table  is  73415. 

The  angle  corresponding  to  this  value  is  10°  27'. 

The  difference  between  73415  and  the  given  mantissa  is  63. 

The  difference  between  73415  and  the  next  following  mantissa  is  71,  and  ff 
of  60"  =  53". 

As  the  function  is  decreasing,  the  53"  must  be  subtracted  from  10°  27';  and 
the  required  angle  is  10°  26'  7". 


Exercises. 
Find 

'     1.  log  sin  30°    8'    9".  9.  log  tan  25°  27' 47' 

2.  log  sin  54°  54' 40".  10.  log  cos  56°  11' 57' 

3.  log  cos  43°  32' 31".  11.  log  cot  62°    0'    4' 

4.  log  cos  69°  25' 11".  12.  log  cos  75°  26' 58' 

5.  log  tan  32°    9'  17".  13.  log  tan  33°  27'  13' 

6.  log  tan  50°    2'    2".  14.  log  cot  81°  55'  24' 

7.  log  cot  44°  33'  17".  15.  log  tan  89°  46'  35' 

8.  log  cot  55°    9' 32".  16.  log  tan    1°  25' 56' 


XVI 


LOGARITHMS. 

i  the  angle  A  if 

17.   log  sin  ^=    9.70075. 

25. 

log  cos  A  = 

9.40008. 

18.   log  sin  ^=    9.91289. 

26. 

log  cot  A  = 

9.78815. 

19.   log  cos  ^=    9.86026. 

27. 

log  cos  A  = 

9.34.301. 

20.    log  cos  ^=    9.54595. 

28. 

log  tan  A  = 

10.52288. 

21.   log  tan  ^=    9.79840. 

29. 

log  cot  A  = 

9  65349. 

22.    log  tan^  =  10.07671. 

30. 

log  sin  ^  = 

8.39316. 

23.    log  cot  ^  =  10.00675. 

31. 

log  sin  A  = 

8.06678. 

24.   log  cot  J.  =    9.84266. 

32. 

log  tan  A  = 

8.11148. 

27.  If  log  sec  or  log  esc  of  an  angle  is  desired,  it  may  be  found 
from  the  table  by  the  formulas, 

sec  A  = 7 ;  hence,  log  sec  A  =  colog  cos  A. 

cos  A 

CSC  A  =  —. — 7 ;  hence,  log  esc  A  ==  colog  sin  A. 
smA 

Page  31.     log  sec    8°  28'        =  colog  cos   8°  28'         =  0.00476. 
Page  42.     log  esc  59°  36'  44"  =  colog  sin  59°  36'  44"  =  0.06418. 

28.  If  a  given  angle  is  between  0°  and  1°,  or  between  89°  and  90°; 
or,  conversely,  if  a  given  log  sin  or  log  cos  does  not  lie  between  the 
limits  8.24186  and  9.99993  in  the  table;  or,  if  a  given  log  tan  or 
log  cot  does  not  lie  between  the  limits  8.24192  and  11.75808  in  the 
table ;  then  pages  21-24  of  Table  III.  must  be  used. 

On  page  21,  log  sin  of  angles  between  0°  and  0°  3',  pr  log  cos  of 
the  complementary  angles  between  89°  57'  and  90°,  are  given  to 
every  second;  for  the  angles  between  0°  and  0°  3',  log  tan  =  log  sin, 
and  log  cos  =  0.00000 ;  for  the  angles  between  89°  5T  and  90°, 
log  cot  =:  log  COS,  and  log  sin  =  0.00000. 

On  pages  22-24,  log  sin,  log  tan,  and  log  cos  of  angles  between 
0°  and  1°,  or  log  cos,  log  cot,  and  log  sin  of  the  complementary 
angles  between  89°  and  90°,  are  given  to  every  10". 

Whenever  log  tan  or  log  cot  is  not  given,  they  may  be  found  by 
the  formulas, 

log  tan  =  colog  cot.  log  cot  =  colog  tan. 

Conversely,  if  a  given  log  tan  or  log  cot  is  not  contained  in  the 
table,  then  the  colog  must  be  found ;  this  will  be  the  log  cot  or 
log  tan,  as  the  case  may  be,  and  will  be  contained  in  the  table. 

On  pages  25-27  the  logarithms  of  the  functions  of  angles 
between  1°  and  2°,  or  between  88°  and  90°,  are  given  in  the  manner 
employed  on  pages  22-24.  These  pages  should  be  used  if  the  angle 
lies  between  these  limits,  and  if  not  only  degrees  and  minutes,  but 
degrees,  minutes,  and  multiples  of  10"  are  given  or  required. 


INTRODUCTION.  XVll 

When  the  angle  is  between  0°  and  2°,  or  88°  and  90°,  and  a 
greater  degree  of  accuracy  is  desired  than  that  given  by  the  table, 
interpolation  may  be  employed ;  but  for  these  angles  interpolation 
does  not  always  give  true  results,  and  it  is  better  to  use  Table  IV. 

Find  log  tan  0°  2'  47",  and  log  cos  89°  37'  20". 

Page  21.      log  tan    0°    2' 47'' =  log  sin  0°  2M7'' =  6.90829  -  10. 
Page  23.      log  cos  89°  37'  20"  =  7.81911  -  10. 

Find  log  cot  0°  2'  15". 

10  -10 

Page  21.       log  tan  0°  2'  15"     =    6.81591  —  10 
Therefore,  log  cot  0°  2' 15"     =    3.18409 

Find  log  tan  89°  38'  30". 

10  - 10 

Page  23.      log  cot  89°  38'  30"  =    7.79617-10 
Therefore,  log  tan  89°  38' 30"  =    2.20383 

Find  the  angle  for  which  log  tan  =  6.92090  — 10. 

Page  21.     The  nearest  log  tan  is  6.92110  —  10. 
The  corresponding  angle  for  which  is  0°  2'  52". 

Find  the  angle  for  which  log  cos  =  7.70240  — 10. 

Page  22.     The  nearest  log  cos  is  7.70261  —  10. 
The  corresponding  angle  for  which  is  89°  42'  40". 

Find  the  angle  for  which  log  cot  =  2.37368. 

This  log  cot  is  not  contained  in  the  table. 
The  colog  cot  =  7.62632  —  10  =  log  tan. 

The  log  tan  in  the  table  nearest  to  this  is  (page  22)  7.62510—10,  and  the 
angle  corresponding  to  this  value  of  log  tan  is  0°  14'  30". 

29.  If  an  angle  x  is  between  90°  and  360°,  it  follows,  from 
formulas  established  in  Trigonometry,  that, 

between  90°  and  180°,  between  180°  and  270°, 

log  sin  X  =  log  sin  (180°  —  x),  log  sin  x  =  log  sin  (x  ~  180°)„, 

log  cos  X  =  log  cos  (180°  —  ic)„,  log  cos  x  =  log  cos  (x  — 180°)„, 

log  tan  X  =  log  tan  (180°  —  x)^,  log  tan  x  =  log  tan  (x  — 180°), 

log  cot  X  =  log  cot  (180°  —  x)^ ;  log  cot  x  =  log  cot  (x  — 180°) ; 

between  270°  and  360°, 
log  sin  X  =  log  sin  (360°  —  x)^^ 
log  cos  £c  =  log  cos  (360°  — ic), 
log  tan  X  =  log  tan  (360°  —  x)„, 
log  cot  ic  =  log  cot  (360°  — x)„. 


XVIU  LOGARITHMS. 

The  letter  n  is  placed  (according  to  custom)  after  the  logarithms 
of  those  functions  which  are  negative  in  value. 

The  above  formulas  show,  without  further  explanation,  how  to 
find  by  means  of  Table  III.  the  logarithms  of  the  functions  of  any 
angle  between  90°  and  360°. 

Thus,  log  sin  137°  45'  22"  =  log  sin  42°  14'  38''  =  9.82766  -  10. 
log  cos  137°  45'  22"  =  log„  cos  42°  14'  38"  =  9.86940„  -  10. 
log  tan  137°  46'  22"  =  log„  tan  42°  14'  38"  =  9.95816n  -  10. 
log  cot  137°  45'  22"  =  log„  cot  42°  14'  38"  =  0.04186„. 
log  sin  209°  32'  60"  =  log„  sin  29°  32'  60"  ==  9.69297„  -  10. 
log  cos  330°  27'  10"  =  log  cos   29°  32'  50"  =  9.93949  -  10. 

Conversely,  to  a  given  logarithm  of  a  trigonometric  function 
there  correspond  between  0°  and  360°  four  angles,  one  angle  in 
each  quadrant,  and  so  related  that  if  x  denote  the  acute  angle,  the 
other  three  angles  are  180°  —  ic,  180°  -|- a;,  and  360°  —  a;. 

If  besides  the  given  logaritlim  it  is  known  whether  the  function 
is  positive  or  negative,  the  ambiguity  is  confined  to  two  quadrants, 
therefore  to  two  angles. 

Thus,  if  the  log  tan  =  9.47451  —  10,  the  angles  are  10°  3G'  17"  in  Quadrant  I. 
and  196°  36'  17"  in  Quadrant  III.;  but  if  the  log  tan  =  9.47451„—  10,  the  angles 
are  163°  23'  43"  m  Quadrant  II.  and  343°  23'  4.3"  in  Quadrant  IV. 

To  remove  all  ambiguity,  further  conditions  are  required,  or  a 
knowledge  of  the  special  circumstances  connected  with  the  problem 
in  question.  •    , 

TABLE  IV.   Q/juOX 

30.  This  table  (page  50)  must  be  used  when  great  accuracy  is 
desired  in  working  with  angles  between  0°  and  2°,  or  between  88° 
and  90°. 

The  values  of  S  and  T  are  such  that  when  the  angle  a  is 
expressed  in  seconds, 

S  =  log  sin  a  —  log  a", 
T  =  log  tan  a  —  log  a". 

Hence  follow  the  formulas  given  on  i)age  50. 

The  values  of  S  and  T  are  printed  with  the  characteristic  10  too 
large,  and  in  using  them  — 10  must  always  be  annexed. 


Find  log  sin  0°  58'  17". 

0°  58'  17"  =  3497" 
log  3497  =  3.64370 

S  =  4.68565  -  10 
log  sin  0°  58'  17"  =  8.22926  -  10 


Find  log  cos  88°  26'  41.2". 

90°  -  88°  26'  41.2"  =  1°  .33'  18.8" 
=  5598.8" 
log  5598.8  =  3.74809 

S  =  4.68562  -  10 
log  cos  88°  20'  41.2"  =  8.43361  -  10 


INTRODUCTION.  XIX 


Find  log  tan  0°  52'  47.5". 

0°  62' 47.5"  =  3167.5" 
log  3167.5  =  3.50072 

T  =  4.68561  -  10 
log  tan  0°  52'  47.5"  =  8. 18633  -  10 


Find  log  tan  89**  54'  37.362". 
90°  —  89°  54'  37.362"  =  0°  5'  22.638" 
=  322.638" 
log  322.638  =  2.50871 

T  =  4.68558  —  10 
log  cot  89°  54'  37.362"  =  7.19420  —  10 
log  tan  89°  54'  37.362"  =  2.80571 


Find  the  angle,  if  log  sin  =  6.72306  — 10. 

6.72306  - 10 
S  =  4.68557  -  10 
Subtract,     2.03749  =  log  109.015 

109.015"        =  0°  1'  49.015". 

Find  the  angle  for  which  log  cot  =  1.67604. 

colog  cot  =  8.32396  —  10 
T  =  4.68564  -  10 
Subtract,     3.63832  =  log  4348.3 

4348.3"  =  1°  12'  28.3". 

Find  the  angle  for  which  log  tan  =  1.55407. 

colog  tan  =  8.44593  —  10 
T  =  4.68569  -  10 
Subtract,     3.76024  =  log  5767.6 

5757.6"  =  1°  35'  67.6", 

and      90°  -  1°  35'  57.6"  =  88°  24'  2.4". 
Therefore,  the  angle  reqmred  is  88°  24'  2.4". 

TABLE   V. 

31.  This  table  (p.  51),  containing  the  circumferences  and  areas 
of  circles,  does  not  require  explanation. 

TABLE   VI. 

82.  Table  VI.  (pp.  52-69)  contains  the  natural  sines,  cosines, 
tangents,  and  cotangents  of  angles  from  0°  to  90°,  at  inter- 
vals of  1'.  If  greater  accuracy  is  desired  it  may  be  obtained 
by  interpolation. 

Note.  In  preparing  the  preceding  explanations,  we  have  made  free  use 
of  the  Logarithmic  Tables  by  F.  G.  Gauss.  For  Table  VI.  we  are  indebted 
to  D.  Carhart 

TABLE    Vn. 

33.  This  table  (pp.  70-75)  gives  the  latitude  and  departure  to 
three  places  of  decimals  for  distances  from  1  to  10,  corresponding 
to  bearings  from  0""  to  90"*  at  intervals  of  15'. 


XX 


LOGARITHMS. 


If  the  bearing  does  not  exceed  45°  it  is  found  in  the  left-h^md 
column,  and  the  designations  of  the  columns  under  "Distance" 
are  taken  from  the  toj)  of  the  page;  but  if  the  bearing  exceeds 
45°,  it  is  found  in  the  right-hsind  column,  and  the  designations 
of  the  columns  under  ^'Distance"  are  taken  from  the  botto7?i  of 
the  page. 

The  method  of  using  the  table  will  be  made  plain  by  the  follow- 
ing examples :  — 

(1)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  N.  35°  15'  E.  6  chains. 

On  p.  75,  left-hand  column,  look  for  35°  15' ;  opposite  this  bearing,  in  the 
vertical  column  headed  "Distance  6,"  are  found  4.900  and  3.463  under  the 
headings  "Latitude"  and  "Departure"  respectively.  Hence,  latitude  or 
northing  =  4.900  chains,  and  departure  or  easting  =  3.463  chains. 

(2)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  S.  87°  W.  2  chains. 

As  the  hearing  exceeds  45°,  we  look  in  the  right-hand  column  of  p.  70,  and 
opposite  87°  in  the  column  marked  "  Distance  2  "  we  find  (taking  the  designa- 
tions of  the  columns  from  the  bottom  of  the  page)  latitude  =  0.105  chains,  and 
departure  =  1.997  chains.  Hence,  latitude  or  southings  0.105  chains,  and 
departure  or  westing  =  1.997  chains. 

(3)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  N.  15°  45'  W.  27.36  chains. 

In  this  case  we  find  the  required  numbers  for  each  figure  of  the  distance 
separately,  arranging  the  work  as  in  the  following  table.  In  practice,  only  the 
last  columns  under  "  Latitude  "  and  "  Departure  "  are  written. 


Distance. 

LAxrruDE. 

Departure. 

20   =  2  X  10 
7 

0.3  =3^10 
0.06  =  6  ^  100 

1.925  X  10  =  19.25 

6.737 

2.887 -MO  =0.289 

5.775 -r  100  =  0.058 

0.543  X  10  =  5.43 
1.90 
0.814^10  =0.081 
1.628  -i- 100  =  0.016 

27.36 

26.334 

7.427 

Hence,  latitude  =  26.334  chains,  and  departure  =  7.427  chains. 


#• 

TABLE  I. 

THE 

COMMON  OE  BRIGGS  LOGAEITHMS 

OF  THE 

NATUEAL  NUMBEES 

From  1  to  10000. 

1-100 

N          log 

N 

log 

N          log 

N 

log 

N          log 

1    0.00  000 

21 

1.32  222 

41     1.  61  278 

61 

1.78  533 

81     1.90  849 

2    0.30103 

22 

1.34  242 

42     1.62  325 

62 

1.  79  239 

82     1.  91  381 

3    0.47  712 

23 

1.  36  173 

43     1.63  347 

63 

1.79  934 

83     1.91908 

4    0.60  206 

24 

1.38  021 

44     1.64  345 

64 

1.80  618 

84     1.92  428 

5    0.69  897 

25 

1.39  794 

45     1.  65  321 

65 

1.  81  291 

85     1.  92  942 

6    0.77  815 

26 

1.  41  497 

46     1.66  276 

66 

1.  81  954 

86    1.93  450 

7    0.84  510 

27 

1.  43  136 

47     1.67  210 

67 

1.  82  607 

87    1.93  952 

8    0.90  309 

28 

1.  44  716 

48     1.68124 

68 

1.  83  251 

88     1.  94  448 

9    0.95  424 

29 

1.46  240 

49     1.69  020 

69 

1.  83  885 

89     1.94  939 

10    1.00  000 

30 

1.47  712 

50     1.69  897 

70 

1.84  510 

90     1.  95  424 

11     1.04  139 

31 

1.  49  136 

51     1.70  757 

71 

1.  85  126 

91     1.  95  904 

12    1.07  918 

32 

1.50  515 

52     1.  71  600 

72 

1.  85  733 

92     1.96  379 

13    1.11394 

33 

1.  51  851 

53     1.72  428 

73 

1.86  332 

93     1.96  848 

14    1.  14  613 

34 

1.  53  148 

54     1.  73  239 

74 

1.86  923 

94     1.97  313 

15    1.17  609 

35 

1.  54  407 

55     1.74  036 

75 

1.  87  506 

95     1.  97  772 

16    1.20  412 

36 

1.55  630 

56     1.74  819 

76 

1.  88  081 

96    1.98  227 

17    1.23  045 

37 

1.56  820 

57    1.75  587 

77 

1.88  649 

97     1.98  677 

18    1.25  527 

38 

1.57  978 

58     1.76  343 

78 

1.  89  209 

98     1.  99  123 

19    1.27  875 

39 

1.  59  106 

59     1.  77  085 

79 

1.  89  763 

99     1.99  564 

20    1.30103 

40 

1.60  206. 

60    1.  77  815 

80 

1.  90  309 

100     2.00  000 

N          log 

N 

log 

N          log 

N 

log 

N          log 

1-100 


100-150 


N 

O 

1 

2    3 

4 

5 

6 

7 

8 

9 

lOO 

00  000 

00  043 

00  087  00130 

00173 

00  217 

00  260 

00  303 

00  346 

00  389 

101 

00  432 

00  475 

00  518  00  561 

00  604 

00  647 

00  689 

00  732 

00  775 

00  817 

102 

00  860 

00  903 

00  945  00  988 

01030 

01072 

01115 

01157 

01199 

01242 

103 

01284 

01326 

01368  01410 

01452 

01494 

01536 

01578 

01620 

01662 

104 

01703 

01745 

01787  01828 

01870 

01912 

01953 

01995 

02  036 

02  078 

105 

02  119 

02160 

02  202  02  243 

02  284 

02  325 

02  366 

02  407 

02  449 

02  490 

106 

02  531 

02  572 

02  612  02  653 

02  694 

02  735 

02  776 

02  816 

02  857 

02  898 

107 

02  938 

02  979 

03  019  03  060 

03  100 

03  141 

03  181 

03  222 

03  262 

03  302 

108 

03  342 

03  383 

03  423  03  463 

03  503 

03  543 

03  583 

03  623 

03  663 

03  703 

109 

03  743 

03  782 

03  822  03  862 

03  902 

03  941 

03  981 

04  021 

04  060 

04100 

llO 

04139 

04  179 

04  218  04  258 

04  297 

04  336 

04  376  04  415 

04  454 

04  493 

111 

04  532 

04  571 

04  610  04  650 

04  689 

04  727 

(H766 

04  805 

04  844 

04  883 

112 

04  922 

04  961 

04  999  05  038 

05  077 

05  115 

05  154 

05  192 

05  231 

05  269 

113 

05  308 

05  346 

05  385  05  423 

05  461 

05  500 

05  538 

05  576 

05  614 

05  652 

114 

05  690 

05  729 

05  767  05  805 

05  843 

05  881 

05  918 

05  956 

05  994 

06  032 

115 

06  070 

06108 

06145  06183 

06  221 

06  258 

06  296 

06  333 

06  371 

06  408 

116 

06  446 

06  483 

06  521  06  558 

06  595 

06  633 

06  670 

06  707 

06  744 

06  781 

117 

06  819 

06  856 

06  893  06  930 

06  967 

07  004 

07  041 

07  078 

07115 

07  151 

118 

07188 

07  225 

07  262  07  298 

07  335 

07  372 

07  408 

07  445 

07  482 

07  518 

119 

07  555 

07  591 

07  628  07  664 

07  700 

07  737 

07.773 

07  809 

07  846 

07  882 

120 

07  918 

07  954 

07  990  08  027 

08  063 

08  099 

08135 

08171 

08  207 

08  243 

121 

08  279 

08  314 

08  350  08  386 

08  422 

08  458 

08  493 

08  529 

08  565 

08  600 

122 

08  636 

08  672 

08  707  08  743 

08  778 

08  814 

08  849 

08  884 

08  920 

08  955 

123 

08  991 

09  026 

09  061  09  096 

09132 

09167 

09  202 

09  237 

09  272 

09  307 

124 

09  342 

09  377 

09  412  09  447 

09  482 

09  517 

09  552 

09  587 

09  621 

09  656 

125 

09  691 

09  726 

09  760  09  795 

09  830 

09  864 

09  899 

09  934 

09  968 

10  003 

126 

10  037 

10  072 

10106  10140 

10175 

10  209 

10  243 

10  278 

10  312 

10  346 

127 

10  380 

10  415 

10  449  10  483 

10  517 

10  551 

10  585 

10  619 

10  653 

10  687 

128 

10  721 

10  755 

10  789  10  823 

10  857 

10  890 

10  924 

10  958 

10  992 

11025 

129 

11059 

11093 

11126  11160 

11193 

11227 

11261 

11294 

11327 

11361 

130 

•11  394 

11428 

11461  11494 

11528 

11561 

11594 

11628 

11661 

11694 

131 

11727 

11760 

11793  11826 

11860 

11893 

11926 

11959 

11  992 

12  024 

132 

12  057 

12  090 

12  123  12156 

12189 

12  222 

12  254 

12  287 

12  320 

12  352 

133 

12  385 

12  418 

12  450  12  483 

12  516 

12  548 

12  581 

12  613 

12  646 

12  678 

134 

12  710 

12  743 

12  775  12  808 

12  840 

12S72 

12  905 

12  937 

12  969 

13  001 

135 

13  033 

13  066 

13  098  13  130 

13  162 

13194 

13  226 

13  258 

13  290 

13  322 

136 

13  354 

13  386 

13  418  13  450 

13  481 

13  5J3 

13  545 

13  577 

13  609 

13  640 

137 

13  672 

13  704 

13  735  13  767 

13  799 

13  830 

13  862 

13  893 

13  925 

13  956 

138 

13  988 

14  019 

14  051  14  082 

14114 

14145 

14176 

14  208 

14  239 

14  270 

139 

14  301 

14  333 

14  364  14  395 

14  426 

14  457 

14  489 

14  520 

14  551 

14  582 

140 

14  613 

14  644 

14  675  14  706 

14  737 

14  768 

14  799 

14  829 

14  860 

14  891 

141 

14  922 

14  953 

14  983  15  014 

15  045 

15  076 

15  106 

15  137 

15  168 

15  198 

142 

15  229 

15  259 

15  290  15  320 

15  351 

15  381 

15  412 

15  442 

15  473 

15  503 

143 

15  534 

15  564 

15  594  15  625 

15  655 

15  685 

15  715 

15  746 

15  776 

15  806 

144 

15  836 

15  866 

15  897  15  927 

15  957 

15  987 

16  017 

16  047 

16  077 

16107 

145 

16137 

16167 

16  197  16  227 

16  256 

16  286 

16  316 

16  346 

16  376 

16  406 

146 

16  435 

16  465 

16  495  16  524 

16  554 

16  584 

16  613 

16  643 

16  673 

16  702 

147 

16  732 

16  761 

16  791  16  820 

16  850 

16  879 

16  909 

16  938 

16  967 

16  997 

148 

17  026 

17  056 

17  085  17114 

17143 

17173 

17  202 

17  231 

17  260 

17  289 

149 

17  319 

17  348 

17  377  17  406 

17  435 

17  464 

17  493 

17  522 

17  551 

17  580 

150 

17  609 

17  638 

17  667  17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

IS^ 

O 

1 

2    3 

4 

5 

6 

7 

8  . 

9 

100-160 


150-200, 

3 

N 

O    1 

2 

-3 

4" 

5 

6 

7 

8 

9 

150 

17  609  17  638 

17  667 

17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

151 

17  898  17  926 

17  955 

17  984 

18  013 

18  041 

18  070 

18  099 

18127 

18156 

152 

18184  18  213 

18  241 

18  270 

18  298 

18  327 

18  355 

18  384 

18  412 

18  441 

153 

18  469  18  498 

18  526 

18  554 

18  583 

18  611 

18  639 

18  667 

18  696 

18  724 

154 

18  752  18  780 

18  808 

18  837 

18  865 

18  893 

18  921 

18  949 

18  977 

19  005 

155 

19  033  19  061 

19  089 

19117 

19145 

19173 

19  201 

19  229 

19  257 

19  285 

156 

19  312  19  340 

19  368 

19  396 

19  424 

19  451 

19  479 

19  507 

19  535 

19  562 

157 

19  590  19  618 

19  645 

19  673 

19  700 

19  728 

19  756 

19  783 

19  811 

19  838 

158 

19  866  19  893 

19  921 

19  948 

19  976 

20  003 

20  030 

20  058 

20  085 

20112 

159 

20140  20167 

20194 

20  222 

20  249 

20  276 

20303 

20  330 

20  358 

20  385 

160 

20  412  20  439 

20  466 

20  493 

20  520 

20  548 

20  575 

20  602 

20  629 

20  656 

161 

20  683  20  710 

20  737 

20  763 

20  790 

20  817 

20  844 

20  871 

20  898 

20  925 

162 

20  952  20  978 

21005 

21032 

21059 

21085 

21112 

21139 

21165 

21192 

163 

21219  21245 

21272 

21299 

21325 

21352 

21378 

21405 

21431 

21458 

164 

21484  21511 

21537 

21564 

21590 

21617 

21643 

21669 

21696 

21722 

165 

21748  21775 

21  801 

21827 

21854 

21880 

21906 

21932 

21958 

21985 

166 

22  011  22  037 

22  063 

22  089 

22  115 

22141 

22  167 

22  194 

22  220 

22  246 

167 

22  272  22  298 

22  324 

22  350 

22  376 

22  401 

22  427 

22  453 

22  479 

22  505 

168 

22  531  22  557 

22  583 

22  608 

22  634 

22  660 

22  686 

22  712 

22  737 

22  763 

169 

22  789  22  814 

22  840 

22  866 

22  891 

22  917 

22  943 

22  968 

22  994 

23  019 

170 

23  045  23  070 

23  096 

23  121 

23147 

23172 

23198 

23  223 

23  249 

23  274 

171 

23  300  23  325 

23  350 

23  376 

23  401 

23  426 

23  452 

23  477 

23  502 

23  528 

172 

23  553  23  578 

23  603 

23  629 

23  654 

23  679 

23  704 

23  729 

23  754 

23  779 

173 

23  805  23  830 

23  855 

23  880 

23  905 

23  930 

23  955 

23  980 

24  005 

24  030 

174 

24  055  24  080 

24105 

24130 

24155 

24180 

24  204 

24  229 

24  254 

24  279 

175 

24  304  24  329 

24  353 

24  378 

24  403 

24  428 

24  452 

24  477 

24  502 

24  527 

176 

24  551  24  576 

24  601 

24  625 

24  650 

24  674 

24  699 

24  724 

24  748 

24  773 

177 

24  797  24  822 

24  846 

24  871 

24  895 

24  920 

24  944 

24  969 

24  993 

25  018 

178 

25  042  25  066 

25  091 

25115 

25139 

25  164 

25  188 

25  212 

25  237 

25  261 

179 

25  285  25  310 

25  334 

25  358 

25  382 

25  406 

25  431 

25  455 

25  479 

25  503  * 

180 

25  527  25  551 

25  575 

25  600 

25  624 

25  648 

25  672 

25  696 

25  720 

25  744 

181 

25  768  25  792 

25  816 

25  840 

25  864 

25  888 

25  912 

25  935 

25  959 

25  983 

182 

26  007  26  031 

26  055 

26  079 

26102 

26126 

26150 

26174 

26198 

26  221 

183 

26  245  26  269 

26  293 

26  316 

26  340 

26  364 

26  387 

26  411 

26  435 

26  458 

184 

26  482  26  505 

26  529 

26  553 

26  576 

26  600 

26  623 

26  647 

26  670 

26  694 

185 

26  717  26  741 

26  764 

26  788 

26  811 

26  834 

26  858 

26  881 

26  905 

26  928 

186 

26  951  26  975 

26  998 

27  021 

27  0+5 

27  068 

27  091 

27  114 

27138 

27  161 

187 

27184  27  207 

27  231 

27  254 

27  277 

27  300 

27  323 

27  346 

27  370 

27  393 

188 

27  416  27  439 

27  462 

27  485 

27  508 

27  531 

27  554 

27  577 

27  600 

27  623 

189 

27  646  27  669 

27  692 

27  715 

27  738 

27  761 

27  784 

27  807 

27  830 

27  852 

190 

27  875  27  898 

27  921 

27  944 

27  967 

27  989 

28  012 

28  035 

28  058 

28  081 

191 

28  103  28126 

28149 

28171 

28194 

28  217 

28  240 

28  262 

28  285 

28  307 

192 

28  330  28  353 

28  375 

28  398 

28  421 

28  443 

28  466 

28  488 

28  511 

28  533 

193 

28  556  28  578 

28  601 

28  623 

28  646 

28  668 

28  691 

28  713 

28  735 

28  758 

194 

28  780  28  803 

28  825 

28  847 

28  870 

28  892 

28  914 

28  937 

28  959 

28  981 

195 

29  003  29  026 

29  048 

29  070 

29  092 

29115 

29137 

29159 

29181 

29  203 

196 

29  226  29  248 

29  270 

29  292 

29  314 

29  336 

29  358 

29  380 

29  403 

29  425 

197 

29  447  29  469 

29  491 

29  513 

29  535 

29  557 

29  579 

29  601 

29  623 

29  645 

198 

29  667  29  688 

29  710 

29  732 

29  754 

29  776 

29  798 

29  820 

29  842 

29  863 

199 

29  885  29  907 

29  929 

29  951 

29  973 

29  994 

30  016 

30  038 

30  060 

30  081 

200 

30103  30125 

30146 

30168 

30190 

30  211 

30  233 

30  255 

30  276 

30  298 

N 

O    1 

2 

3 

4 

5 

6 

7 

8 

9 

160  -  200 


200-250 


N 

O    1 

2 

3 

4 

5 

6    7    8    9 

200 

30103  30125 

30146 

30168 

30190 

30  211 

30  233  30  255  30  276  30  298 

201 

30  320  30  341 

30  363 

30  384 

30  406 

30  428 

30  449  30  471  30  492  30  514 

202 

30  535  30  557 

30  578 

30  600 

30  621 

30  643 

30  664  30  685  30  707  30  728 

203 

30  750  30  771 

30  792 

30  814 

30  835 

30  856 

30  878  30  899  30  920  30  942 

204 

30  963  30  984 

31006 

31027 

31048 

31069 

31091  31112  31133  31154 

205 

31175  31197 

31218 

31239 

31260 

31281 

31302  31323  31345  31366 

206 

31387  31408 

31429 

31450 

31471 

31492 

31513  31534  31555  31576 

207 

31597  31618 

31639 

31660 

31681 

31702 

31  723  31  744  31  765  31  785 

208 

31806  31827 

31848 

31869 

31890 

31911 

31931  31952  31973  31994 

209 

32  015  32  035 

32  056 

32  077 

32  098 

32118 

32139  32160  32181  32  201 

210 

32  222  32  243 

32  263 

32  284 

32  305 

32  325 

32  346  32  366  32  387  32  408 

211 

32  428  32  449 

32  469 

32  490 

32  510 

32  531 

32  552  32  572  32  593  32  613 

212 

32  634  32  654 

32  675 

32  695 

32  715 

32  736 

32  756  32  777  32  797  32  818 

213 

32  838  32  858 

32  879 

32  899 

32  919 

32  940 

32  960  32  980  33  001  33  021 

214 

33  041  33  062 

33  082 

33102 

33  122 

33  143 

33163  33183  33  203  33  224 

215 

33  244  33  264 

33  284 

33  304 

33  325 

33  345 

33  365  33  385  33  405  33  425 

216 

33  445  33  465 

33  486 

33  506 

33  526 

33  546 

33  566  33  586  33  606  33  626 

217 

33  646  33  666 

33  686 

33  706 

33  726 

33  746 

33  766  33  786  33  806  33  826 

218 

33  846  33  866 

33  885 

33  905 

33  925 

33  945 

33  965  33  985  34  005  34  025 

219 

34  044  34  064 

34  084 

34104 

34  124 

34143 

34163  34183  34  203  34  223 

220 

34  242  34  262 

34  282 

34  301 

34  321 

34  341 

34  361  34  380  34  400  34  420 

221 

34  439  34  459 

34  479 

34  498 

34  518 

34  537 

34  557  34  577  34  596  34  616 

222 

34  635  34  655 

34  674 

34  694 

34  713 

34  733 

34  753  34  772  34  792  34  811 

223 

34  830  34  850  34  869  34  889  34  908 

34  928 

34  947  34  967  34  986  35  005 

224 

35  025  35  044 

35  064 

35  083 

35102 

35  122 

35141  35  160  35  180  35  199 

225 

35  218  35  238 

35  257 

35  276 

35  295 

35  315 

35  334  35  353  35  372  35  392 

226 

35  411  35  430 

35  449 

35  468 

35  488 

35  507 

35  526  35  545  35  564  35  583 

227 

35  603  35  622 

35  641 

35  660 

35  679 

35  698 

35  717  35  736  35  755  35  774 

228 

35  793  35  813 

35  832 

35  851 

35  870 

35  889 

35  908  35  927  35  946  35  96i 

229 

35  984  36  003 

36  021 

36  040 

36  059 

36  078 

36  097  36116  36135  36154 

230 

36173  36192 

36  211 

36  229 

36  248 

36  267 

36  286  36  305  36  324  36  342 

231 

36  361  36  380 

36  399 

36  418 

36  436 

36  455 

36  474  36  493  36  511  36  530 

232 

36  549  36  568 

36  586 

36  605 

36  624 

36  642 

36  661  36  680  36  698  36  717 

233 

36  736  36  754 

36  773 

36  791 

36  810 

36  829 

36  847  36  866  36  884  36  903 

234 

36  922  36  940 

36  959 

36  977 

36  996 

37  014 

37  033  37J251  37  070  37  088 
37  218  37  236  37  254  37  273 

235 

37107  37125 

37144 

37162 

37181 

37199 

236 

37  291  37  310 

37  328 

37  346 

37  365 

37  383 

37  401  37  420  37  438  37  457 

237 

37  475  37  493 

37  511 

37  530 

37  548 

37  566  37  585  37  603  37  621  37  639  | 

238 

37  658  37  676 

37  694 

37  712 

37  731 

37  749 

37  767  37  785  37  803  37  822 

239 

37  840  37  858 

37  876 

37  894 

37  912 

37  931 

37  949  37  967  37  985  38  003 

240 

38  021  38  039 

38  057 

38075 

38  093 

38112 

38130  38148  38166  38184 

241 

38  202  38  220 

38  238 

38  256 

38  274 

38  292 

38  310  38  328  38  346  38  364 

242 

38  382  38  399 

38  417 

38  435 

38  453 

38  471 

38  489  38  507  38  525  38  543 

243 

38  561  38  578 

38  596 

38  614 

38  632 

38  650 

38  668  38  686  38  703  38  721 

244 

38  739  38  757 

38  775 

38  792 

38  810 

38  828 

38  846  38  863  38  881  38  899 

245 

38  917  38  934 

38  952 

38  970 

38  987 

39  005 

39  023  39  041  39  058  39  076 

246 

39  094  39111 

39129 

39146 

39164 

39182 

39199  39  217  39  235  39  252 

247 

39  270  39  287 

39  305 

39  322 

39  340 

39  358 

39  375  39  393  39  410  39  428 

248 

39  445  39  463 

39  480 

39  498 

39  515 

39  533 

39  550  39  568  39  585  39  602 

.249 

39  620  39  637 

39  655 

39  672 

39  690 

39  707 

39  724  39  742  39  759  39  777 

250 

39  794  39  811 

39  829 

39  846 

39  863 

39  881 

39  898  39  915  39  933  39  950 

N 

O    1 

2 

3  - 

■  4 

5 

6    7    8    9 

200-260 


260-300 


N 

0    12    3    4 

5    6 

7 

8    9 

250 

39  794  39  811  39  829  39  846  39  863 

39  881  39  898 

39  915 

39  933  39  950 

251 

39  967  39  985  40  002  40  019  40  037 

40  054  40  071 

40088 

40106  40123 

252 

40140  40  157  40  175  40192  40  209 

40  226  40  243 

40  261 

40  278  40  295 

253 

40  312  40  329  40  346  40  364  40  381 

40  398  40  415 

40  432 

40449  40466 

254 

40  483  40  500  40  518  40  535  40  552 

40  569  40  586 

40  603 

40  620  40637 

255 

40  654  40  671  40  688  40  705  40  722 

40  739  40  756 

40  773 

40  790  40807 

256 

40  824  40  841  40  858  40  875  40  892 

40  909  40  926 

40  943 

40  960  40976 

257 

40  993  41010  41027  41044  41061 

41078  41095 

41111 

41  128  41  145 

258 

41162  41179  41196  41212  41229 

41246  41263 

41280 

41296  41313 

259 

41330  41347  41363  41380  41397 

41414  41430 

41447 

41464  41481 

260 

41497  41514  41531  41547  41564 

41581  41597 

41614 

41631  41647 

261 

41664  41681  41697  41714  41731 

41747  41764 

41780 

41797  41814 

262 

41830  41847  41863  41880  41896 

41913  41929 

41946 

41963  41979 

263 

41996  42  012  42  029  42  045  42  062 

42  078  42  095 

42  111 

42127  42144 

264 

42  160  42  177  42  193  42  210  42  226 

42  243  42  259 

42  275 

42  292  42  308 

265 

42  325  42  341  42  357  42  374  42  390 

42  406  42423 

42  439 

42  455  42  472 

266 

42  488  42  504  42  521  42  537  42  553 

42  570  42  586 

42  602 

42  619  42  635 

267 

42  651  42  667  42  684  42  700  42  716 

42  732  42  749 

42  765 

42  781  42  797 

268 

42  813  42  830  42  846  42  862  42  878 

42  894  42  911 

42  927 

42  943  42  959 

269 

42  975  42  991  43  008  43  024  43  040 

43  056  43  072 

43  088 

43  104  43120 

270 

43  136  43  152  43  169  43  185  43  201 

43  217  43  233 

43  249 

43  265  43  281 

271 

43  297  43  313  43  329  43  345  43  361 

43  377  43  393 

43  409 

43  425  43  441 

272 

43  457  43  473  43  489  43  50i  43  521 

43  537  43  553 

43  569 

43  584  43  600 

273 

43  616  43  632  43  648  43  664  43  680 

43  696  43  712 

43  727 

43  743  43  759 

274 

43  775  43  791  43  807  43  823  43  838 

43  854  43  870 

43  886 

43  902  43  917 

275 

43  933  43  949  43  965  43  981  43  996 

44  012  44  028 

44  044 

44  059  44  075 

276 

44  091  44107  44  122  44  138  44  154 

44170  44185 

44  201 

44  217  44  232 

277 

44  248  44  264  44  279  44  295  44  311 

44  326  44  342 

44  358 

44  373  44  389 

278 

44  404  44  420  44  436  44  451  44  467 

44  483  44  498 

44  514 

44  529  44  545 

279 

44  560  44  576  44  592  44  607  44  623 

'44  638  4i^654  44  669  44  685  44  700  | 

280 

44  716  44  731  44  747  44  762  44  778 

44  793  44  809 

44  824 

44  840  44  855 

281 

44  871  44  886  44  902  44  917  44  932 

44  948  44  963 

44  979 

44  994  45  010 

282 

45  025  45  040  45  056  45  071  45  086 

45  102  45  117 

45  133 

45  148  45  163 

283 

45  179  45  194  45  209  45  225  45  240 

45  255  45  271 

45  286 

45  301  45  317 

284 

45  332  45  347  45  362  45  378  45  393 

45  408  45  423 

45  439 

45  454  45  469 

285 

45  484  45  500  45  515  45  530  45  545 

45  561  45  576 

45  591 

45  606  45  621 

286 

45  637  45  652  45  667  45  682  45  697 

45  712  45  728 

45  743 

45  758  45  773 

287 

45  788  45  803  45  818  45  834  45  849 

45  864  45  879 

45  894 

45  909  45  924 

288 

45  939  45  954  45  969  45  984  46  000 

46  015  46  030 

46  045 

46  060  46  075 

289 

46  090  46105  46120  46135  46150 

46165  46180  46  19i 

46  210  46  225 

290 

46  240  46  255  46  270  46  285  46  300 

46  315  46  330 

46  34i 

46  359  46  374 

291 

46  389  46  404  46  419  46  434  46  449 

46  464  46  479 

46  494 

46  509  46  523 

292 

46  538  46  553  46  568  46  583  46  598 

46  613  46  627 

46  642 

46  657  46  672 

293 

46  687  46  702  46  716  46  731  46  746 

46  761  46  776 

46  790 

46  805  46  820 

294 

46  835  46  850  46  864  46  879  46  894 

46  909  46  923 

46  938 

46  953  46  967 

295 

46  982  46  997  47  012  47  026  47  041 

47  056  47  070 

47  085 

47100  47114 

296 

47  129  47144  47  159  47173  47  188 

'  47  202  47  217 

47  232 

47  246  47  261 

297 

47  276  47  290  47  305  47  319  47  334  \ 

47  349  47  363 

47  378 

47  392  47  407 

298 

47  422  47  436  47  451  47  465  47  480 

47  494  47  509 

47  524 

47  538  47  553 

299 

47  567  47  582  47  596  47  611  47  625 

47  640  47  654 

47  669 

47  683  47  698 

300 

47  712  47  727  47  741  47  756  47  770 

47  784  47  799 

47  813 

47  828  47  842 

N 

O    1    2    3    4 

5    6 

7 

8    9 

250-300 


/'' 


300-360 


f^f-VV»" 


N 

O 

1 

2 

3    4  . 

5 

6 

7 

8 

9 

300 

47  712 

47  727 

47  741 

47  756  47  770 

47  784 

47  799 

47  813 

47  828 

47  842 

301 

47  857 

47  871 

47  885 

47  900  47  914 

47  929 

47  943 

47  958 

47  972 

47  986 

302 

48  001 

48  015 

48  029 

48  044  48  058 

48  073 

48  087 

48101 

48116 

48130 

303 

48144 

48159 

48173 

48187  48  202 

48  216 

48  230 

48  244 

48  259 

48  273 

304 

48  287 

48  302 

48  316 

48  330  48  344 

48  359 

48  373 

48  387 

48  401 

48  416 

305 

48  430 

48  444 

48  458 

48  473  48  487 

48  501 

48  515 

48  530 

48  544 

48  558 

306 

48  572 

48  586 

48  601 

48  615  48  629 

48  643 

48  657 

48  671 

48  686 

48  700 

307 

48  714 

48  728 

48  742 

48  756  48  770 

48  785 

48  799 

48  813 

48  827 

48  841 

308 

48  855 

48  869 

48  883 

48  897  48  911 

48  926 

48  940 

48  954 

48  968 

48  982 

309 

48  996 

49  010 

49  024 

49  038  49  052 

49  066 

49  080 

49  094 

49108 

49122 

310 

49136 

49150 

49164 

49178  49192 

49  206 

49  220 

49  234 

49  248 

49  262 

311 

49  276 

49  290 

49  304 

49  318  49  332 

49  346 

49  360 

49  374 

49  388 

49  402 

312 

49  415 

49  429 

49  443 

49  457  49  471 

49  485 

49  499 

49  513 

49  527 

49  541 

313 

49  554 

49  568 

49  582 

49  596'  49  610 

49  624 

49  638 

49  651 

49  665 

49  679 

314 

49  693 

49  707 

49  721 

49  734  49  748 

49  762 

49  776 

49  790 

49  803 

49  817 

315 

49  831 

49  845 

49  859 

49  872  49  886 

49  900 

49  914 

49  927 

49  941 

49  955 

316 

49  969 

49  982 

49  996 

50  010  50  024 

50  037 

50  051 

50  065 

50  079 

50  092 

317 

50106 

50120 

50133 

50147  50161 

50174 

50188 

50  202 

50  215 

50  229 

318 

50  243 

50  256 

50  270 

50  284  50  297 

50  311 

50  325 

50  338 

50  352 

50  365 

319 

50  379 

50  393 

50  406 

50  420  50  433 

50  447 

50  461 

50  474 

50  488 

50  501 

320 

50  515 

50  529 

50  542 

50  556  50  569 

50  583 

50  596 

50  610 

50  623 

50  637 

321 

50  651 

50  664 

50  678 

50  691  50  705 

50  718 

50  732 

50  745 

50  759 

50  772 

322 

50  786 

50  799 

50  813 

50  826  50  840 

50  853 

50  866 

50  880 

50  893 

50  907 

323 

50  920 

50  934 

50  947 

50  961  50  974 

50  987 

51001 

51014 

51  028 

51041 

324 

51055 

51068 

51081 

51095  5110S 

51121 

51135 

51148 

51162 

51175 

325 

51188 

51202 

51215 

51228  51242 

51  255 

51268 

51282 

51  295 

51  308 

326 

51322 

51335 

51348 

51362  51375 

51388 

51402 

51415 

51428 

51441 

327 

51455 

51468 

51481 

51495  51508 

51  521 

51534 

51548 

51561 

51574 

328 

51587 

51601 

51614 

51627  51640 

51654 

51667 

51  680 

51  693 

51706 

329 

51720 

51733 

51746 

51759  51772 

51786 

51799 

51812 

51825 

51838 

330 

51851 

51865 

51  878 

51891  51904 

51  917 

51930 

51943 

51  957 

51970 

331 

51  983 

51996 

52  009 

52  022  52  035 

52  048 

52  061 

52  075 

52  088 

52101 

332 

52  114 

52  127 

52140 

52  153  52166 

52  179 

52  192 

52  205 

52  218 

52  231 

333 

52  244 

52  257 

52  270 

52  284  52  297 

52  310 

52  323 

52  336 

52  349 

52  362 

334 

52  375 

52  388 

52  401 

52  414  52  427 

52  440 

52  453 

52  466 

52  479 

52  492 

335 

52  504 

52  517 

52  530 

52  543  52  556 

52  569 

52  582 

52  595 

52  608 

52  621 

336 

52  634 

52  647 

52  660 

52  673  52  686 

52  699 

52  711 

52  724 

52  737 

52  750 

337 

52  763 

52  776 

52  789 

52  802  52  815 

52  827 

52  840 

52  853 

52  866 

52  879 

338 

52  892 

52  905 

52  917 

52  930  52  943 

52  956 

52  969 

52  982 

52  994 

53  007 

339 

53  020 

53  033 

53  046 

53  058  53  071 

53  084 

53  097 

53  110 

53  122 

53  135 

340 

53  148 

53  161 

53  173 

53  186  53  199 

53  212 

53  224 

53  237 

53  250 

53  263 

341 

53  275 

53  288 

53  301 

53  314  53  326 

53  339 

53  352 

53  364 

53  377 

53  390 

342 

53  403 

53  415 

53  428 

53  441  53  453 

53  466 

53  479 

53  491 

53  504 

53  517 

343 

53  529 

53  542 

53  555 

53  567  53  580 

53  593 

53  605 

53  618 

53  631 

53  643 

344 

53  656 

53  668 

53  681 

53  694  53  706 

53  719 

53  732 

53  744 

53  757 

53  769 

345 

53  782 

53  794 

53  807 

53  820  53  832 

53  845 

53  857 

53  870 

53  882 

53  895 

346 

53  908 

53  920 

53  933 

53  945  53  958 

53  970 

53  983 

53  995 

54  008 

54  020 

347 

54  033 

54  045 

54  058 

54  070  54  083 

54  095 

54  108 

54  120 

54  133 

54145 

348 

54  158 

54  170 

54183 

54195  54  208 

54  220 

54  233 

54  245 

54  258 

54  270 

349 

54  283 

54  295 

54  307 

54  320  54  332 

54  345 

54  357 

54  370 

54  382 

54  394 

350 

54  407 

54  419 

54  432 

54  444  54  456 

54  469 

54  481 

54  494 

54  506 

54  518 

N 

O 

1 

2 

3    4 

5 

6 

7 

8 

9 

300-360 


35€>-400  m  ; 

7 

N 

O 

1 

2 

3 

4 

-T       6 

7 

8 

9 

350 

54  407 

54  419 

54  432 

54  444 

54  456 

54469  54  481 

54  494 

54  506 

54  518 

351 

54  531 

54  543 

54  555 

54  568 

54  580 

54  593  54  605 

54  617 

54  630 

54  642 

352 

54  654 

54  667 

54  679 

54  691 

54  704 

54  716  54  728 

54  741 

54  753 

54  765 

353 

54  777 

54  790 

54  802 

54  814 

54  827 

54  839  54  851 

54  864 

54  876 

54  888 

354 

54  900 

54  913 

54  925 

54  937 

54  949 

54  962  54  974 

54  986 

54  998 

55  011 

355 

55  023 

55  035 

55  047 

55  060 

55  072 

55  084  55  096 

55  108 

55  121 

55133 

356 

55  145 

55  157 

55  169 

55  182 

55  194 

55  206  55  218 

55  230 

55  242 

55  255 

357 

55  267 

55  279 

55  291 

55  303 

55  315 

55  328  55  340 

55  352 

55  364 

55  376 

358 

55  388 

55  400 

55  413 

55  425 

55  437 

55  449  55  461 

55  473 

55  485 

55  497 

359 

55  509 

55  522 

55  534 

55  546 

55  558 

55  570  55  582 

55  594 

55  606 

55  618 

360 

55  630 

55  642 

55  654 

55  666 

55  678 

55  691  55  703 

55  715 

55  727 

55  739 

361 

55  751 

55  763 

55  775 

55  787 

55  799 

55  811  55  823 

55  835 

55  847 

55  859 

362 

55  871 

55  883 

55  895 

55  907 

55  919 

55  931  55  943 

55  955 

55  967 

55  979 

363 

55  991 

56  003 

56  015 

56  027 

56  038 

56  050  56  062 

56  074 

56  086 

56  098 

364 

56110 

56122 

56134 

56146 

56158 

56170  56182 

56194 

56  205 

56  217 

365 

56  229 

56  241 

56  253 

56  265 

56  277 

56  289  56  301 

56B12 

56  324 

56  336 

366 

56  348 

56  360 

56  372 

56  384 

56  396 

56  407  56  419 

56  431 

56  443 

56  455 

367 

56  467 

56  478 

56  490 

56  502 

56  514 

56  526  56  538 

56  549 

56  561 

56  573 

368 

56  585 

56  597 

56  608 

56  620 

56  632 

56  644  56  656 

56  667 

56  679 

56  691 

369 

56  703 

56  714 

56  726 

56  738 

56  750 

56  761  56  773 

56  785 

56  797 

56  808 

370 

56  820 

56  832 

56  844 

56  855 

56  867 

56  879  56  891 

56  902 

56  914 

56  926 

371 

56  937 

56  949 

56  961 

56  972 

56  984 

56996  57  008 

57  019 

57  031 

57  043 

372 

57  054 

57  066 

57  078 

57  089 

57101 

57113  57,124 

57136 

57  148 

57159 

373 

57171 

57  183 

57  194 

57  206 

57  217 

57  229  57  241 

57  252 

57  264 

57  276 

374 

57  287 

57  299 

57  310 

57  322 

57  334 

57  345  57  357 

57  368 

57  380 

57  392 

375 

57  403 

57  415 

57  426 

57  438 

57  449 

57  461  57  473 

57  484 

57  496 

57  507 

376 

57  519 

57  530 

57  542 

57  553 

57  565 

57  576  57  588 

57  600 

57  611 

57  623 

377 

57  634 

57  646 

57  657 

57  669 

57  680 

57  692  57  703 

57  715 

57  726 

57  738. 

378 

57  749 

57  761 

57  772 

57  784 

57  795 

57  807  57  818 

57  830 

57  841 

57852 

379 

57  864 

57  875 

57  887 

57  898 

57  910 

57  921  57  933 

57  944 

57  955 

57  967 

380 

57  978 

57  990 

58  001 

58  013 

58  024 

58  035  58  047 

58  058 

58  070 

58  081 

381 

58  092 

58104 

58  115 

58127 

58138 

58149  58161 

58172 

58184 

58195 

382 

58  206 

58  218 

58  229 

58  240 

58  252 

58  263  58  274 

58  286 

58  297 

58  309 

383 

58  320 

58  331 

58  343 

58  354 

58  365 

58  377  58  388 

58  399 

58  410 

58  422 

384 

58  433 

58  444 

58  456 

58  467 

58  478 

58  490  58  501 

58  512 

58  524 

58  535 

385 

58  546 

58  557 

58  569 

58  580 

58  591 

58  602  58  614 

58  625 

58  636 

58  647 

386 

58  659 

58  670 

58  681 

58  692 

58  704 

58  715  58  726 

58  737 

58  749 

58  760 

387 

58  771 

58  782 

58  794 

58  805 

58  816 

58  827  58  838 

58  850 

58  861 

58  872 

388 

58  883 

58  894 

58  906 

58  917 

58  928 

58  939  58  950 

58  961 

58  973 

58  984 

389 

58  995 

59  006 

59  017 

59  028 

59  040 

59  051  59  062 

59073 

59  084 

59  095 

390 

59106 

59118 

59129 

59140 

59151 

59162  59173 

59184 

59195 

59  207 

391 

59  218 

59  229 

59  240 

59  251 

59  262 

59  273  59  284 

59  295 

59  306 

59  318 

392 

59  329 

59  340 

59  351 

59  362 

59  373 

59  384  59  395 

59  406 

59  417 

59  428 

393 

59  439 

59  450 

59  461 

59  472 

59  483 

59  494  59  506 

59  517 

59  528 

59  539 

394 

59  550 

59  561 

59  572 

59  583 

59  594 

59  605  59  616 

59  627 

59  638 

59  649 

395 

59  660 

59  671 

59  682 

59  693 

59  704 

59  715  59  726 

59  737 

59  748 

59  759 

396 

59  770 

59  780 

59  791 

59  802 

59  813 

59  824  59  835 

59  846 

59  857 

59  868 

397 

59  879 

59  890 

59  901 

59  912 

59  923 

59  934  59  945 

59  956 

59  966 

59  977 

398 

59  988 

59  999 

60  010 

60  021 

60  032 

60  043  60  054 

60  065 

60  076 

60  086 

399 

60  097 

60108 

60119 

60130 

60141 

60152  60163 

60173 

60184 

60195 

400 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260  60  271 

60  282 

60  293 

60  304 

N 

O 

1 

2 

3 

4 

5    6 

7 

8 

9 

360-400 


400-460 


N 

O    1    2    3    4 

5    6    7    8    9 

400 

60  206  60  217  60  228  60  239  60  249 

60  260  60  271  60  282  60  293  60  301 

401 

60  314  60  325  60  336  60  347  60  358 

60  369  60  379  60  390  60  401  60  412 

402 

60  423  60  433  60  444  60  455  60  466 

60  477  60  487  60  498  60  509  60  520 

403 

60  531  60  541  60  552  60  563  60  574 

60  584  60  595  60  606  60  617  60  627 

404 

60  638  60  649  60  660  60  670  60  681 

60  692  60  703  60  713  60  724  60  73i 

405 

60  746  60  756  60  767  60778  60  788 

60  799  60  810  60  821  60  831  60  842 

406 

60  853  60  863  60  874  60  885  60  895 

60  906  60  917  60  927  60  938  60  949 

407 

60  959  60  970  60  981  60  991  61002 

61013  61023  61034  61045  61055 

408 

61066  61077  61087  61098  61109 

61  119  61  130  61  140  61  151  61  162 

409 

61172  61183  61194  61204  61215 

61225  61236  61247  61257  61268 

410 

61278  61289  61300  61310  61321 

61331  61342  61352  61363  61374 

411 

61384  61395  61405  61416  61426 

61437  61448  61458  61469  61479 

412 

61490  61500  61511  61521  61532 

61542  61553  61563  61574  61584 

413 

61595  61606  61616  61627  61637 

61648  61658  61669  61679  61690 

414 

61700  61711  61721  61731  61742 

61752  61763  61773  61784  61794 

415 

61805  61815  61826  61836  61847 

61857  61868  61878  61888  61899 

416 

61909  61920  61930  61941  61951 

61962  61972  61 982  61993  62  003 

417 

62  014  62  024  62  034  62  045  62  055 

62  066  62  076  62  086  62  097  62  107 

418 

62118  62  128  62  138  62  149  62  159 

62  170  62  180  62  190  62  201  62  211 

419 

62  221  62  232  62  242  62  252  62  263 

62  273  62  284  62  294  62  304  62  315 

420 

62  325  62  335  62  346  62  356  62  366 

62  377  62  387  62  397  62  408  62  418 

421 

62  428  62  439  62  449  62  459  62  469 

62  480  62  490  62  500  62  511  62  521 

422 

62  531  62  542  62  552  62  562  62  572 

62  583  62  593  62  603  62  613  62  624 

423 

62  634  62  644  62  655  62  665  62  675 

62  685  62  696  62  706  62  716  62  726 

424 

62  737  62  747  62  757  62  767  62  778 

62  788  62  798  62  808  62  818  62  829 

425 

62  839  62  849  62  859  62  870  62  880 

62  890  62  900  62  910  62  921  62  931 

426 

62  941  62  951  62  961  62  972  62  982 

62  992  63  002  63  012  63  022  63  033 

427 

63  043  63  053  63  063  63  073  63  083 

63  094  63  104  63  114  63  124  63  134 

428 

63  144  63  155  63  165  63  175  63  185 

63  195  63  205  63  215  63  225  63  236 

429 

63  246  63  256  63  266  63  276  63  286 

63  296  63  306  63  317  63  327  63  337 

430 

63  347  63  357  63  367  63  377  63  387 

63  397  63  407  63  417  63  428  63  438 

431 

63  448  63  458  63  468  63  478  63  488 

63  498  63  508  63  518  63  528  63  538 

432 

63  548  63  558  63  568  63  579  63  589 

63  599  63  609  63  619  63  629  63  639 

433 

63  649  63  659  63  669  63  679  63  689 

63  699  63  709  63  719  63  729  63  739 

434 

63  749  63  759  63  769  63  779  63  789 

63  799  63  809  63  819  63  829  63  839 

435 

63  849  63  859  63  869  63  879  63  889 

63  899  63  909  63  919  63  929  63  939 

436 

63  949  63  959  63  969  63  979  63  988 

63  998  64  008  64  018  64  028  64  038 

437 

64  048  64  058  64  068  64  078  64  088 

64  098  64  108  64  118  64  128  ^4 137 

438 

64  147  64  157  64167  64177  64187 

64  197  64  207  64  217  64  227  64  237 

439 

64  246  64  256  64  266  64  276  64  286 

64  296  64  306  64  316  64  326  64  335 

440 

64  345  64  355  64  365  64  375  64  385 

64  395  64  404  64  414  64  424  64  434 

441 

64  444  64  454  64  464  64  473  64  483 

64  493  64  503  64  513  64  523  64  532 

442 

64  542  64  552  64  562  64  572  64  582 

64  591  64  601  64  611  64  621  64  631 

443 

64  640  64  650  64  660  64  670  64  680 

64  689  64  699  64  709  64  719  64  729 

444 

64  738  64  748  64  758  64  768  64  777 

64  787  64  797  64  807  64  816  64  826 

445 

64  836  64  846  64  856  64  865  64  875 

64  885  64  895  64  904  64  914  64  924 

446 

64  933  64  943  64  953  64  963  64  972 

64  982  64  992  65  002  65  011  65  021 

447 

65  031  65  040  65  050  65  060  65  070 

65  079  65  089  65  099  65  108  65  118 

448 

65  128  65  137  65  147  65  157  65  167 

65  176  65  186  65  196  65  205  65  215 

449 

65  225  65  234  65  244  65  254  65  263 

65  273  65  283  65  292  65  302  65  312 

450 

65  321  65  331  65  341  65  350  65  360 

65  369  65  379  65  389  65  398  65  408 

N 

O    1    2    3    4 

5    6    7    8    9 

400-460 


46Q,-600 

* 

9 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321 

65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408 

451 

65  418 

65  427 

65  437 

65  447 

65  456 

65  466 

65  475 

65  485 

65  495 

65  504 

452 

65  514 

65  523 

65  533 

65  543 

65  552 

65  562 

65  571 

65  581 

65  591 

65  600 

453 

65  610 

65  619 

65  629 

65  639 

65  648 

65  658 

65  667 

65  677 

65  686 

65  696 

454 

65  706 

65  715 

65  725 

65  734 

65  744 

65  753 

65  763 

65  772 

65  782 

65  792 

455 

65  801 

65  811 

65  820 

65  830 

65  839 

65  849 

65  858 

65  868 

65  877 

65  887 

456 

65  896 

65  906 

65  916 

65  925 

65  935 

65  944 

65  954 

65  963 

65  973 

65  982 

457 

65  992 

66  001 

66  011 

66  020 

66  030 

66  039 

66  049 

66  058 

66  068 

66  077 

458 

66  087 

66  096 

66106 

66115 

66124 

66134 

66143 

66153 

66162 

66172 

459 

66181 

66191 

66  200 

66  210 

66  219 

66  229 

66  238 

66  247 

66  257 

66  266 

460 

66  276 

66  285 

66  295 

66  304 

66  314 

66  323 

66  332 

66  342 

66  351 

66  361 

461 

66  370 

66  380 

66  389 

66  398 

66  408 

66  417 

66  427 

66  436 

66  445 

66  455 

462 

66  464 

66  474 

66  483 

66  492 

66  502 

66  511 

66  521 

66  530 

66  539 

66  549 

463 

66  558 

66  567 

66  577 

66  586 

66  596 

66  605 

66  614 

66  624 

66  633 

66  642 

464 

66  652 

66  661 

66  671 

66  680 

66  689 

66  699 

66  708 

66  717 

66  727 

66  736 

465 

66  745 

66  755 

66  764 

66  773 

66  783 

66  792 

66  801 

66  811 

66  820 

66  829 

466 

66  839 

66  848 

66  857 

66  867 

66  876 

66  885 

66  894 

66  904 

66  913 

66  922 

467 

66  932 

66  941 

66  950 

66  960 

66  969 

66  978 

66  987 

66  997 

67  006 

67  015 

468 

67  025 

67  034 

67  043 

67  052 

67  062 

67  071 

67  080 

67  089 

67  099 

67  108 

469 

67  117 

67  127 

67136 

67  145 

67154 

67164 

67173 

67]  82 

67191 

67  201 

470 

67  210 

67  219 

67  228 

67  237 

67  247 

67  256 

67  265 

67  274 

67  284 

67  293 

471 

67  302 

67  311 

67  321 

67  330 

67  339 

67  348 

67  357 

67  367 

67  376 

67  385 

472 

67  394 

67  403 

67  413 

67  422 

67  431 

67  440 

67  449 

67  459 

67  468 

67  477 

473 

67  486 

67  495 

67  504 

67  514 

67  523 

67  532 

67  541 

67  550 

67  560 

67  569 

474 

67  578 

67  587 

67  596 

67  605 

67  614 

67  624 

67  633 

67  642 

67  651 

67  660 

475 

67  669 

67  679 

67  688 

67  697 

67  706 

67  715 

67  724 

67  733 

67  742 

67  752 

476 

67  761 

67  770 

67  779 

67  788 

67  797 

67  806 

67  815 

67  825 

67  834 

67  843 

477 

67  852 

67  861 

67  870 

67  879 

67  888 

67  897 

67  906  67  916  67  925 

67  934 

478 

67  943 

67  952 

67  961 

67  970 

67  979 

67  988 

67  997 

68  006 

68  015 

68  024 

479 

68  034 

68  043 

68  052 

68  061 

68  070 

68  079 

68  088 

68  097 

68106 

68115 

480 

68124 

68133 

68142 

68151 

68160 

68169 

68178 

68187 

68196 

68  205 

481 

68  215 

68  224 

68  233 

68  242 

68  251 

68  260 

68  269 

68  278 

68  287 

68  296 

482 

68  305 

68  314 

68  323 

68  332 

68  341 

68  350 

68  359 

68  368 

68  377 

68  386 

483 

68  395 

68  404 

68  413 

68  422 

68  431 

68  440 

68  449 

68  458 

68  467 

68  476 

484 

68  485 

68  494 

68  502 

68  511 

68  520 

68  529 

68  538 

68  547 

68  556 

68  565 

485 

68  574 

68  583 

68  592 

68  601 

68  610 

68  619 

68  628 

68  637 

68  646 

68  655 

486 

68  664 

68  673 

68  681 

68  690 

68  699 

68  708 

68  717 

68  726 

68  735 

68  744 

487 

68  753 

68  762 

68  771 

68  780 

68  789 

68  797 

68  806 

68  815 

68  824 

68  833 

488 

68  842 

68  851 

68  860 

68  869 

68  878 

68  886 

68  895 

68  904 

68  913 

68  922 

489 

68  931 

68  940 

68  949 

68  958 

68  966 

68  975 

68  984 

68  993 

69  002 

69  011 

490 

69  020 

69  028 

69  037 

69  046 

69  055 

69  064 

69  073 

69  082 

69  090 

69099 

491 

69108 

69117 

69126 

69135 

69  144 

69152 

69161 

69170 

69  179 

69188 

492 

69197 

69  205 

69  214 

69  223 

69  232 

69  241 

69  249 

69  258 

69  267 

69  276 

493 

69  285 

69  294 

69  302 

69  311 

69  320 

69  329 

69  338 

69  346 

69  355 

69  364 

494 

69  373 

69  381 

69  390 

69  399 

69  408 

69  417 

69  425 

69  434 

69  443 

69  452 

495 

69  461 

69  469 

69  478 

69  487 

69  496 

69  504 

69  513 

69  522 

69  531 

69  539 

496 

69  548 

69  557 

69  566 

69  574 

69  583 

69  592 

69  601 

69  609 

69  618 

69  627 

497 

69  636 

69  644 

69  653 

69  662 

69  671 

69  679 

69  688 

69  697 

69  705 

69  714 

498 

69  723 

69  732 

69  740 

69  749 

69  758 

69  767 

69  775 

69  784 

69  793 

69  801 

499 

69  810 

69  819 

69  827 

69  836 

69  845 

69  854 

69  862 

69  871 

69  880 

69  888 

500 

69  897 

69  906 

69  914 

69  923 

69  932 

69  940 

69  949 

69  958 

69  966 
8 

69  975 

N 

O 

1 

2 

3 

4 

5 

6 

7 

9 

460-600 


10 


50a-660 


N 

O 

1 

2 

3 

4 

5 

6 

7    8 

9 

500 

69  897 

69  906 

69  914 

69  923 

69  932 

69  940 

69  949 

69  958  69  966 

69  975 

501 

69  984 

69  992 

70  001 

70  010 

70  018 

70  027 

70  036 

70  044  70  053 

70  062 

502 

70  070 

70  079 

70  088 

70  096 

70105 

70114 

70122 

70131  70140 

70148 

503 

70157 

70165 

70174 

70183 

70  191 

70  200 

70  209 

70  217  70  226 

70  234 

504 

70  243 

70  252 

70  260 

70  269 

70  278 

70  286 

70  295 

70  303  70  312 

70  321 

505 

70  329 

70  338 

70  346 

70  355 

70  364 

70  372 

70  381 

70  389  70  398 

70  406 

506 

70  415 

70  424 

70  432 

70441 

70  449 

70  458 

70  467 

70  475  70  484 

70  492 

507 

70  501 

70  509 

70  518 

70  526 

70  535 

70  544 

70  552 

70  561  70  569 

70  578 

508 

70  586 

70  595 

70  603 

70  612 

70  621 

70  629 

70  638 

70  646  70  655 

70  663 

509 

70  672 

70  680 

70  689 

70  697 

70  706 

70  714 

70  723 

70  731  70  740 

70  749 

510 

70  757 

70  766 

70  774 

70  783 

70  791 

70  800 

70  808 

70  817  70  825 

70  834 

511 

70  842 

70  851 

70  859 

70  868 

70  876 

70  885 

70  893 

70  902  70  910 

70  919 

512 

70  927 

70  935 

70  944 

70  952 

70  961 

70  969 

70  978 

70  986  70  995 

71003 

513 

71012 

71020 

71  029 

71037 

71046 

71  054 

71063 

71071  71079 

71088 

514 

71096 

71105 

71  113 

71122 

71130 

71139 

71147 

71155  71164 

71172 

515 

71181 

71189 

71198 

71206 

71214 

71223 

71231 

71240  71248 

71257 

516 

71265 

71273 

71282 

71290 

71299 

71307 

71  315 

71324  71332 

71341 

517 

71349 

71357 

71366 

71374 

71383 

71391 

71399 

71408  71416 

71425 

518 

71433 

71441 

71450 

71458 

71466 

71475 

71483 

71492  71500 

71  508 

519 

71517 

71525 

71533 

71542 

71550 

71559 

71567 

71575  71584 

71592 

520 

71600 

71609 

71617 

71625 

71634 

71642 

71650 

71659  71667 

71675 

521 

71684 

71692 

71700 

71709 

71717 

71725 

71734 

71742  71750 

71759 

522 

71767 

71775 

71784 

71792 

71800 

71809 

71817 

71825  71834 

71842 

523 

71  850 

71  858 

71867 

71875 

71883 

71892 

71900 

71908  71917 

71925 

524 

71933 

71941 

71950 

71958 

71  966 

71975 

71983 

71991  71999 

72  008 

525 

72  016 

72  024 

72  032 

72  041 

72  049 

72  057 

72  066 

72  074  72  082 

72  090 

526 

72  099 

72  107 

72  115 

72  123 

72  132 

72  140 

72  148 

72  156  72  165 

72  173 

527 

72  181 

72  189 

72198 

72  206 

72  214 

72  222 

72  230 

72  239  72  247 

72  255 

528 

72  263 

72  272 

72  280 

72  288 

72  296 

72  304 

72  313 

72  321  72  329 

72  337 

529 

72  346 

72  354 

72  362 

72  370 

72  378 

72  387 

72  395 

72  403  72  411 

72  419 

530 

72  428 

72  436 

72  444 

72  452 

72  460 

72  469 

72  477 

72  485  72  493 

72  501 

531 

72  509 

72  518 

72  526 

72  534 

72  542 

72  550 

72  558 

72  567  72  575 

72  583 

532 

72  591 

72  599 

72  607 

72  616 

72  624 

72  632 

72  640 

72  648  72  656 

72  665 

533 

72  673 

72  681 

72  689 

72  697 

72  705 

72  713 

72  722 

72  730  72  738 

72  746 

534 

72  754 

72  762 

72  770 

72  779 

72  787 

72  795 

72  803 

72  811  72  819 

72  827 

535 

72  835 

72  843 

72  852 

72  860 

72  868 

72  876 

72  884 

72  892  72  900 

72  908 

536 

72  916 

72  925 

72  933 

72  941 

72  949 

72  957 

72  965 

72  973  72  981 

72  989 

537 

72  997 

73  006 

73  014 

73  022 

73  030 

73  038 

73  046 

73  054 '73  062 

73  070 

538 

73  078 

73  086 

73  094 

73  102 

73  111 

73  119 

73  127 

73  135  73  143 

73  151 

539 

73  159 

73  167 

73175 

73  183 

73  191 

73  199 

73  207 

73  215  73  223 

73  231 

540 

73  239 

73  247 

73  255 

73  263 

73  272 

73  280 

.73  288 

73  296  73  304 

73  312 

541 

73  320 

73  328 

73  336 

73  344 

73  352 

73  360 

73  368 

73  376  73  384 

73  392 

542 

73  400 

73  408 

73  416 

73  424 

73  432 

73  440 

73  448 

73  456  73  464 

73  472 

543 

73  480 

73  488 

73  496 

73  504 

•73  512 

73  520 

73  528 

73  536  73  544 

73  552 

544 

73  560 

73  568 

73  576 

73  584 

73  592 

73  600 

73  608 

73  616  73  624 

73  632 

545 

73  640 

73  648 

73  656 

73  664 

73  672 

73  679 

73  687 

73  695  73  703 

73  711 

546 

73  719 

73  727 

73  735 

73  743 

73  751 

73  759 

73  767 

73  775  73  783 

73  791 

547 

73  799 

73  8C?7 

73  815 

73  823 

73  830 

73  838 

73  846 

73  854  73  862 

73  870 

548 

73.878 

73  886 

73  894 

73  902 

73  910 

73  918 

73  926 

73  933  73  941 

73  949 

549 

73  957 

73  965 

73  973 

73  981 

73  989 

73  997 

74  005 

74  013  74  020 

74  028 

550 

74  036 

74  044 

74  052 

74  060 

74  068 

74  076 

74  084 

74  092  74  099 

74107 

N 

O 

1 

2 

3 

4 

5 

6 

7    8 

9 

500-560 


560  -^PO 

11 

N 

O 

74  036 

1 

2 

3 

4 

5 

6 

7 

8 

74  099 

9 

550 

74  044 

74  052 

74  060 

74  068 

74  07^  74  084 

74  092 

74107 

551 

74115 

74  123 

74  131 

74  139 

74  147 

74  155 

74162 

74  170 

74178 

74186 

552 

74194 

74  202 

74  210 

74  218 

74  225 

74  233 

74  241 

74  249 

74  257 

74  265 

553 

74  273 

74  280 

74  288 

74  296 

74  304 

74  312 

74  320 

74  327 

74  335 

74  343 

554 

74  351 

74  359 

74  367 

74  374 

74  382 

74  390 

74  398 

74  406 

74  414 

74  421 

555 

74  429 

74  437 

74  445 

74  453 

74  461 

74  468 

74  476 

74  484 

74  492 

74^00 

556 

74  507 

74  515 

74  523 

74  531 

74  539 

74  547 

74  554 

74  562 

74  570 

74  578 

557 

74  586 

74  593 

74  601 

74  609 

74  617 

74  624 

74  632 

74  640 

74  648 

74  656 

558 

74  663 

74  671 

74  679 

74  687 

74  695 

74  702 

74  710 

74  718 

74  726 

74  733 

559 

74  741 

74  749 

74  757 

74  764 

74  772 

74  780 

74  788 

74  796 

74  803 

74  811 

560 

74  819 

74  827 

74  834 

74  842 

74  850 

74  858 

74  865 

74  873 

74  881 

74  889 

561 

74  896 

74  904 

74  912 

74  920 

74  927 

74  935 

74  943 

74  950 

74  958 

74  966 

562 

74  974 

74  981 

74  989 

74  997 

75  005 

75  012 

75  020 

75  028 

75  035 

75  043 

563 

75  051 

75  059 

75  066 

75  074 

75  082 

75  089 

75  097 

75  105 

75  113 

75  120 

564 

75  128 

75  136 

75143 

75  151 

75  159 

75  166 

75  174 

75  182 

75  189 

75  197 

565 

75  205 

75  213 

75  220 

75  228 

75  236 

75  243 

75  251 

75  259 

75  266 

75  274 

566 

75  282 

75  289 

75  297 

75  305 

75  312 

75  320 

75  328 

75  335 

75  343 

75  351 

567 

75  358 

75  366 

75  374 

75  381 

75  389 

75  397 

75  404 

75  412 

75  420 

75  427 

568 

75  435 

75  442 

75  450 

75  458 

75  465 

75  473 

75  481 

75  488 

75  496 

75  504 

569 

75  511 

75  519 

75  526 

75  534 

75  542 

75  549 

75  557 

75  565 

75  572 

75  580 

570 

75  587 

75  595 

75  603 

75  610 

75  618 

75  626 

75  633 

75  641 

75  648 

75  656 

571 

75  664 

75  671 

75  679 

75  686 

75  694 

75  702 

75  709 

75  717 

75  724 

75  732 

572 

75  740 

75  747 

75  755 

75  762 

75  770 

75  778 

75  785 

75  793 

75  800 

75  808 

573 

75  815 

75  823 

75  831 

75  838 

75  846 

75  853 

75  861 

75  868 

75  876 

75  884 

574 

75  891 

75  899 

75  906 

75  914 

75  921 

75  929 

75  937 

75  944 

75  952 

75  959 

575 

75  967 

75  974 

75  982 

75  989 

75  997 

76  005 

76  012 

76  020 

76  027 

76  035 

576 

76  042 

76  050 

76  057 

76  065 

76  072 

76  080 

76  087 

76  095 

76103 

76110 

577 

76118 

76125 

76133 

76140 

76148 

76155 

76163 

76170 

76178 

76185 

578 

76193 

76  200 

76  208 

76  215 

76  223 

76  230 

76  238 

76  245 

76  253 

76  260 

579 

76  268 

76  275 

76  283 

76  290 

76  298 

76  305 

76  313 

76  320 

76  328 

76  335 

580 

76  343 

76  350 

76  358 

76  365 

76  373 

76  380 

76  388 

76  395 

76  403 

76  410 

581 

76  418 

76  425 

76  433 

76  440 

76  448 

76455 

76  462 

76  470 

76  477 

76  485 

582 

76  492 

76  500 

76  507 

76  515 

76  522 

76  530 

76  537 

76  545 

76  552 

76  559 

583 

76  567 

76  574 

76  582 

76  589 

76  597 

76  604 

76  612 

76  619 

76  626 

76  634 

584 

76  641 

76  649 

76  656 

76  664 

76  671 

76  678 

76  686 

76  693 

76  701 

76  708 

585 

76  716 

76  723 

76  730 

76  738 

76  745 

76  753 

76  760 

76  768 

76  775 

76  782 

586 

76  790 

76  797 

76  805 

76  812 

76  819 

76  827 

76  834 

76  842 

76  849 

76  856 

587 

76  864 

76  871 

76  879 

76  886 

76  893 

76  901 

76  908 

76  916 

76  923 

76  930 

588 

76  938 

76  945 

76  953 

76  960 

76  967 

76  975 

76  982 

76  989 

76  997 

77  004 

589 

77  012 

77  019 

77  026 

77  034 

77  041 

77  048 

77  056 

77  063 

77  070 

77  078 

590 

77  085 

77  093 

77100 

77  107 

77115 

77122 

77129 

77137 

77144 

77151 

591 

77  159 

77166 

77173 

77181 

77188 

77195 

77  203 

77  210 

77  217 

77  225 

59-2 

^  77  232 

77  240 

77  247 

77  254 

77  262 

77  269 

77  276 

77  283 

77  291 

77  298 

593 

77  305 

77  313 

77  320 

77  327 

77  335 

77  342 

77  349 

77  357 

77  364 

77  371 

594 

77  379 

77  386 

77  393 

77  401 

77  408 

77  415 

77  422 

77  430 

77  437 

77  444 

595 

77  452 

77  459 

77  466 

77  474 

77  481 

77  488 

77  495 

77  503 

77  510 

77  517 

596 

77  525 

77  532 

77  539 

77  546 

77  554 

77  561 

77  568 

77  576 

77  583 

77  590 

597 

77  597 

77  605 

77  612 

77  619 

77  627 

77  634 

77  641 

77  648 

77  656 

77  663 

598 

77  670 

77  677 

77  685 

77  692 

77  699 

77-706 

77  714 

77  721 

77  728 

77  735 

599 

77-743 

77  750 

77  757 

77  764 

77  772 

77  779 

77  786 

77  793 

77  801 

77  808 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851 

77  859 

77  866 

77  873 

77  880 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

860-600 


12 

600-650 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8    9 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851 

77  859 

77  866 

77  873  77  880 

601 

77  887 

77  895 

77  902 

77  909 

77  916 

77  924 

77  931 

77  938 

77  945  77  952 

602 

77  960 

77  967 

77  974 

77  981 

77  988 

77  996 

78  003 

78  010 

78  017  78  025 

603 

78  032 

78  039 

78  046 

78  053 

78  061 

78  068 

78  075 

78  082 

78  089  78  097 

604 

78104 

78111 

78118 

78125 

78132 

78140 

78147 

78154 

78161  78168 

605 

78176 

78183 

78190 

78197 

78  204 

78  211 

78  219 

78  226 

78  233  78  240 

606 

78  247 

78  254 

78  262 

78  269 

78  276 

78  283 

78  290 

78  297 

78  305  78  312 

607 

78  319 

78  326 

78  333 

78  340 

78  347 

78  355 

78  362 

78  369 

78  376  78  383 

608 

78  390 

78  398 

78  405 

78  412 

78  419 

78  426 

78  433 

78  440 

78  447  78  455 

609 

78  462 

78  469 

78  476 

78  483 

78  490 

78  497 

78  504 

78  512 

78  519  78  526 

610 

78  533 

78  540 

78  547 

78  554 

78  561 

78  569 

78  576 

78  583 

78  590  78  597 

611 

78  604 

78  611 

78  618 

78  625 

78  633 

78  640 

78  647 

78  654 

78  661  78  668 

612 

78  675 

78  682 

78  689 

78  696 

78  704 

78  711 

78  718 

78  725 

78  732  78  739 

613 

78  746 

78  753 

78  760 

78  767 

78  774 

78  781 

78  789 

78  796 

78  803  78  810 

614 

78  817 

78  824 

78  831 

78  838 

78  845 

78  852 

78  859 

78  866 

78  873  78  880 

615 

78  888 

78  895 

78  902 

78  909 

78  916 

78  923 

78  930 

78  937 

78  944  78  951 

616 

78  958 

78  965 

78  972 

78  979 

78  986 

78  993 

79  000 

79  007 

79  014  79  021 

617 

79  029 

79  036 

79  043 

79  050 

79  057 

79  064 

79  071 

79  078 

79  085  79  092 

618 

79  099 

79106 

79113 

79120 

79127 

79134 

79141 

79148 

79155  79162 

619 

79169 

79176 

79183 

79190 

79197 

79  204 

79  211 

79  218 

79  225  79  232 

620 

79  239 

79  246 

79  253 

79  260 

79  267 

79  274 

79  281 

79  288 

79  295  79  302 

621 

79  309 

79  316 

79  323 

79  330 

79  337 

79  344 

79  351 

79  358 

79  365  79  372 

622 

79  379 

79  386 

79  393 

79  400 

79  407 

79  414 

79  421 

79  428 

79  435  79  442 

623 

79  449 

79  456 

79  463 

79  470 

79  477 

79  484 

79  491 

79  498 

79  505  79  511 

624 

79  518 

79  525 

79  532 

79  539 

79  546 

79  553 

79  560 

79  567 

79  574  79  581 

625 

79  588 

79  595 

79  602 

79  609 

79  616 

79  623 

79  630 

79  637 

79  644  79  650 

626 

79  657 

79  664 

79  671 

79  678 

79  685 

79  692 

79  699 

79  706 

79  713  79  720 

627 

79  727 

79  734 

79  741 

79  748 

79  754 

79  761 

79  768 

79  775 

79  782  79  789 

628 

79  796 

79  803 

79  810 

79  817 

79  824 

79  831 

79  837 

79  844 

79  851  79  858 

629 

79  865 

79  872 

79  879 

79  886 

79  893 

79  900 

79  906 

79  913 

79  920  79  927 

630 

79  934 

79  941 

79  948 

79  955 

79  962 

79  969 

79  975 

79  982 

79  989  79  996 

631 

80  003 

80  010 

80  017 

80  024 

80  030 

80  037 

80  044 

80  051 

80  058  80  065 

632 

80  072 

80  079 

80  085 

80  092 

80  099 

80106 

80113 

80120 

80127  80134 

633 

80140 

80147 

80154 

80161 

80168 

80175 

80182 

80188 

80195  80  202 

634 

80  209 

80  216 

80  223 

80  229 

80  236 

80  243 

80  250 

80  257 

80  264  80  271 

635 

80  277 

80  284 

80  291 

80  298 

80  305 

80  312 

80  318 

80  325 

80  332  80  339 

636 

80  346 

80  353 

80  359 

80  366 

80  373 

80  380 

80  387 

80  393 

80  400  80  407 

637 

80  414 

80  421 

80  428 

80  434 

80  441 

80  448 

80  455 

80  462 

80  468  80  475 

638 

80  482 

80  489 

80  496 

80  502 

80  509 

80  516 

80  523 

80  530 

80  536  80  543 

639 

80  550 

80  557 

80  564 

80  570 

80  577 

80  584 

80  591 

80  598 

80  604  80  6U 

640 

80  618 

80  625 

80  632 

80  638 

80  645 

80  652 

80  659 

80  665 

80  672  80  579 

641 

80  686 

80  693 

80  699 

80  706 

80  713 

80  720 

80  726 

80  733 

80  740  8^747 

642 

80  754 

80  760 

80767 

80  774 

80  781 

80  787 

80  794 

80  801 

80  808  -^,'.814 

643 

80  821 

80  828 

80  835 

80  841 

80  848 

80  855 

80  862 

80  868 

80  875  80  882 

644 

80  889 

80  895 

80  902 

80  909 

80  916 

80  922 

80  929 

80  936 

80  943  80  949 

645 

80  956 

80  963 

80  969 

80  976 

80  983 

80  990 

80  996 

81003 

81010  81017 

646 

81023 

81030 

81037 

81043 

81050 

81057 

81064 

81070 

81077  81084 

647 

81090 

81097 

81104 

81111 

81117 

81124 

81131 

81137 

81144  81151 

648 

81158 

81164 

81171 

81178 

81184 

81191 

81198 

81204 

81211  81218 

649 

81224 

81231 

81238 

81245 

81251 

81258 

81265 

81  271 

81278  81285 

650 

81291 

81298 

81305 

81311 

81318 

*81  325 

81331 

81338 

81345  81351 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8    9 

600-660 


660-700 

13 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

650 

81291 

81298 

81305 

81311 

81318 

81325 

81331 

81338 

8134i 

81351 

651 

81358 

81365 

81371 

81378 

81385 

81391 

81398 

81405 

81411 

81418 

652 

81425 

81431 

81438 

81445 

81451 

81458 

81465 

81471 

81478 

81485 

653 

81491 

81498 

81  505 

81511 

81518 

8152i 

81531 

81538 

81544 

81551 

654 

81558 

81564 

81571 

81578 

81584 

81591 

81598 

81604 

81611 

81617 

655 

81624 

81631 

81637 

81644 

81651 

81657 

81664 

81671 

81677 

81684 

656 

81690 

81697 

81704 

81710 

81717 

81723 

81730 

81737 

81743 

81750 

657 

81757 

81763 

81770 

81776 

81783 

81790 

81796 

81803 

81809 

81816 

658 

81823 

81829 

81836 

81842 

81849 

81856 

81862 

81869 

81875 

81882 

659 

81889 

81895 

81902 

81908 

81915 

81921 

81928 

81935 

81941 

81948 

660 

81954 

81961 

81968 

81974 

81981 

81987 

81994 

82  000 

82  007 

82  014 

661 

82  020 

82  027 

82  033 

82  040 

82  046 

82  053 

82  060 

82  066 

82  073 

82  079 

662 

82  086 

82  092 

82  099 

82105 

82  112 

82119 

82125 

82132 

82  138 

82  145 

663 

82151 

82  158 

82  164 

82  171 

82178 

82184 

82191 

82197 

82  204 

82  210 

664 

82  217 

82  223 

82  230 

82  236 

82  243 

82  249 

^82  256 

82  263 

82  269 

82  276 

665 

82  282 

82  289 

82  295 

82  302 

82  308 

82  315 

82  321 

82  328 

82  334 

82  341 

666 

82  347 

82  354 

82  360 

82  367 

82  373 

82  380 

82  387 

82  393 

82  400 

82  406 

667 

82  413 

82  419 

82  426 

82  432 

82  439 

82  445 

82  452 

82  458 

82  465 

82  471 

668 

82  478 

82  484 

82  491 

82  497 

82  504 

82  510 

82  517 

82  523 

82  530 

82  536 

669 

82  543 

82  549 

82  556 

82  562 

82  569 

82  575 

82  582 

82  588 

82  595 

82  601 

670 

82  607 

82  614 

82  620 

82  627 

82  633 

82  640 

82  646 

82  653 

82  659 

82  666 

671 

82  672 

82  679 

82  685 

82  692 

82  698 

82  705 

82  711 

82  718 

82  724 

82  730 

672 

82  737 

82  743 

82  750 

82  756 

82  763 

82  769 

82  776 

82  782 

82  789 

82  795 

673 

82  802 

82  808 

82  814 

82  821 

82  827 

82  834 

82  840 

82  847 

82  853 

82  860 

674 

82  866 

82  872 

82  879 

82  885 

82  892 

82  898 

82  90i 

82  911 

82  918 

82  924 

675 

82  930 

82  937 

82  943 

82  950 

82  956 

82  963 

82  969 

82  975 

82  982 

82  988 

676 

82  995 

83  001 

83  008 

83  014 

83  020 

83  027 

83  033 

83  040 

83  046 

83  052 

677 

83  059 

83  065 

83  072 

83  078 

83  085 

83  091 

83  097 

83  104 

83  110 

83117 

678 

83  123 

83  129 

83  136 

83  142 

83  149 

83  155 

83  161 

83  168 

83  174 

83  181 

679 

83  187 

83  193 

83  200 

83  206 

83  213 

83  219 

83  225 

83  232 

83  238 

83  24i 

680 

83  251 

83  257 

83  264 

83  270 

83  276 

83  283 

83  289 

83  296 

83  302 

83  308 

681 

83  315 

83  321 

83  327 

83  334 

83  340 

83  347 

83  353 

83  359 

83  366 

83  372 

^682 

83  378 

83  385 

83  391 

83  398 

83  404 

83  410 

83  417 

83  423 

83  429 

83  436 

683 

83  442 

83  448 

83  455 

83  461 

83  467 

83  474 

83  480 

83  487 

83  493 

83  499 

684 

83  506 

83  512 

83  518 

83  525 

83  531 

83  537 

83  544 

83  550 

83  556 

83  563 

685 

83  569 

83  575 

83  582 

83  588 

83  594 

83  601 

83  607 

83  613 

83  620 

83  626 

686 

83  632 

83  639 

83  645 

83  651 

83  658 

83  664 

83  670 

83  677 

83  683 

83  689 

687 

83  696 

83  702 

83  708 

83  715 

83  721 

83  727 

83  734 

83  740 

83  746 

83  753 

688 

83  759 

83  765 

83  771 

83  778 

83  784 

83  790 

83  797 

83  803 

83  809 

83  816 

689 

83  822 

83  828 

83  835 

83  841 

83  847 

83  853 

83  860 

83  866 

83  872 

83  879 

690 

83  885 

83  891 

83  897 

83  904 

83  910 

83  916 

83  923 

83  929 

83  935 

83  942 

691 

83  948 

83  954 

83  960 

83  967 

83  973 

83  979 

83  985 

83  992 

83  998 

84  004 

692 

84  011 

84  017 

84  023 

84  029 

84  036 

84  042 

84  048 

84  055 

84  061 

84  067 

693 

84  073 

84  080 

84  086 

84  092 

84  098 

84105 

84111 

84117 

84123 

84130 

694 

84  136 

84142 

84148 

84  155 

84  161 

84167 

84173 

84180 

84186 

84192 

695 

84198 

84  205 

84  211 

84  217 

84  223 

84  230 

84  236 

84  242 

84  248 

84  255 

696 

84  261 

84  267 

84  273 

84  280 

84  286 

84  292 

84  298 

84  305 

84  311 

84  317 

697 

84  323 

84  330 

84  336 

84  342 

84  348 

84  354 

84  361 

84  367 

84  373 

84  379 

698 

84  386 

84  392 

84  398 

84  404 

84  410 

84  417 

84  423 

84  429 

84  435 

84  442 

699 

84  448 

84  454 

84  460 

84  466 

84  473 

84  479 

84  485 

84  491 

84  497 

84  504 

TOO 

84  510 

84  516 

84  522 

84  528 

84  535 

84  541 

84  547 

84  553 

84  559 

84  566 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

660-700 


14 

700-760 

N 

O    1    2    3    4 

5    6    7    8    9 

700 

84  510  84  516  84  522  84  528  84  535 

84  541  84  547  84  553  84  559  84  566 

701 

84  572  84  578  84  584  84  590  84  597 

84  603  84  609  84  615  84  621  84  628 

702 

84  634  84  640  84  646  84  652  84  658 

84  665  84  671  84  677  84  683  84  689 

703 

84  696  84  702  84  708  84  714  84  720 

84  726  84  733  84  739  84  745  84  751 

704 

84  757  84  763  84  770  84  776  84  782 

84  788  84  794  84  800  84  807  84  813 

705 

84  819  84  825  84  831  84  837  84  844 

84  850  84  856  84  862  84  868  84  874 

706 

84  880  84  887  84  893  84  899  84  905 

84  911  84  917  84  924  84  930  84  936 

707 

84  942  84  948  84  954  84  960  84  967 

84  973  84  979  84  985  84  991  84  997 

708 

85  003  85  009  85  016  85  022  85  028 

85  034  85  040  85  046  85  052  85  058 

709 

85  065  85  071  85  077  85  083  85  089 

85  095  85  101  85  107  85  114  85  120 

710 

85  126  85132  85  138  85  144  85  150 

85156  85  163  85  169  85  175  85  181 

711 

85  187  85  193  85  199  85  205  85  211 

85  217  85  224  85  230  85  236  85  242 

712 

85  248  85  254  85  260  85  266  85  272 

85  278  85  285  85  291  85  297  85  303 

713 

85  309  85  315  85  321  85  327  85  333 

85  339  85  345  85  352  85  358  85  364 

714 

85  370  85  376  85  382  85  388  85  394 

85  400  85  406  85  412  85  418  85  425 

715 

85  431  85  437  85  443  85  449  85  455 

85  461  85  467  85  473  85  479  85  485 

716 

85  491  85  497  85  503  85  509  85  516 

85  522  85  528  85  534  85  540  85  546 

717 

85  552  85  558  85  564  85  570  85  576 

85  582  85  588  85  594  85  600  85  606 

718 

85  612  85  618  85  625  85  631  85  637 

85  643  85  649  85  655  85  661  85  667 

719 

85  673  85  679  85  685  85  691  85  697 

85  703  85  709  85  715  85  721  85  727 

720 

85  733  85  739  85  745  85  751  85  757 

85  763  85  769  85  775  85  781  85  788 

721 

85  794  85  800  85  806  85  812  85  818 

85  824  85  830  85  836  85  842  85  848 

722 

85  854  85  860  85  866  85  872  85  878 

85  884  85  890  85  896  85  902  85  908 

723 

85  914  85  920  85  926  85  932  85  938 

85  944  85  950  85  956  85  962  85  968 

724 

85  974  85  980  85  986  85  992  85  998 

86  004  86  010  86  016  86  022  86  028 

725 

86  034  86  040  86  046  86  052  86  058 

86  064  86  070  86  076  86  082  86  088 

726 

86  094  86100  86106  86112  86118 

86124  86130  86136  86  141  86147 

727 

86153  86159  86  165  86171  86177 

86183  86189  86  195  86  201  86  207 

728 

86  213  86  219  86  225  86  231  86  237 

86  243  86  249  86  255  86  261  86  267 

729 

86  273  86  279  86  285  86  291  86  297 

86  303  86  308  86  314  86  320  86  326 

730 

86  332  86  338  86  344  86  350  86  356 

86  362  86  368  86  374  86  380  86  386 

731 

86  392  86  398  86  404  86  410  86  415 

86  421  86  427  86  433  86  439  86  445 

732 

86  451  86  457  86  463  86  469  86  475 

86  481  86  487  86  493  86  499  86  504 

733 

86  510  86  516  86  522  86  528  86  534 

86  540  86  546  86  552  86  558  86  564 

734 

86  570  86  576  86  581  86  587  86  593 

86  599  86  605  86  611  86  617  86  623 

735 

86  629  86  635  86  641  86  646  86  652 

86  658  86  664  86  670  86  676  86  682 

736 

86  688  86  694  86  700  86  705  86  711 

86  717  86  723  86  729  86  735  86  741 

737 

86  747  86  753  86  759  86  764  86  770 

86  776  86  782  86  788  86  794  86  800 

738 

86  806  86  812  86  817  86  823  86  829 

86  835  86  841  86  847  86  853  86  859 

739 

86  864  86  870  86  876  86  882  86  888 

86  894  86  900  86  906  86  911  86  917 

740 

86  923  86  929  86  935  86  941  86  947 

86  953  86  958  86  964  86  970  86  976 

741 

86  982  86  988  86  994  86  999  87  005 

87  011  87  017  87  023  87  029  87  035 

742 

87  040  87  046  87  052  87  058  87  064 

87  070  87  075  87  081  87  087  87  093 

743 

87  099  87105  87111  87116  87122 

87  128  87  134  87  140  87  146  87151 

744 

87157  87163  87169  87175  87181 

87186  87192  87198  87  204  87  210 

745 

87  216  87  221  87  227  87  233  87  239 

87  245  87  251  87  256  87  262  87  268 

746 

87  274  87  280  87  286  87  291  87  297 

87  303  87  309  87  315  87  320  87  326 

747 

87  332  87  338  87  344  87  349  87  355 

87  361  87  367  87  373  87  379  87  384 

748 

87  390  87  396  87  402  87  408  87  413 

87  419  87  425  87  431  87  437  87  442 

749 

87  448  87  454  87  460  87  466  87  471 

87  477  87  483  87  489  87  495  87  500 

750 

87  506  87  512  87  518  87  523  87  529 

87  535  87  541  87  547  87  552  87  558 

N 

O    1    2    3    4 

5    6    7    8    9 

700-7^ 

50 

760-800 


15 


N 

O 

1 

87  512 

2 

3 

4 

5 

6 

7 

8 

87  552 

9 

750 

87  506 

87  518 

87  523 

87  529 

87  535 

87  541 

87  547 

87  558 

751 

87  564 

87  570 

87  576 

87  581 

87  587 

87  593 

87  599 

87  604 

87  610 

87  616 

752 

87  622 

87  628 

87  633 

87  639 

87  645 

87  651 

87  656 

87  662 

87  668 

87  674 

753 

87  679 

87  685 

87  691 

87  697 

87  703 

87  708 

87  714 

87  720 

87  726 

87  731 

754 

87  737 

87  743 

87  749 

87  754 

87  760 

87  766 

87  772 

87  777 

87  783 

87  789 

755 

87  795 

87  800 

87  806 

87  812 

87  818 

87  823 

87  829 

87  835 

87  841 

87  846 

756 

87  852 

87  858 

87  864 

87  869 

87  875 

87  881 

87  887 

87  892 

87  898 

87  904 

757 

87  910 

87  915 

87  921 

87  927 

87  933 

87  938 

87  944 

87  950 

87  955 

87  961 

758 

87  967 

87  973 

87  978 

87  984 

87  990 

87  996 

88  001 

88  007 

88  013 

88  018 

759 

88  024 

88  030 

88  036 

88  041 

88  047 

88  053 

88  058 

88  064 

88  070 

88  076 

760 

88  081 

88  087 

88  093 

88  098 

88104 

88110 

88116 

88121 

88127 

88133 

761 

88138 

88144 

88150 

88156 

88161 

88167 

88173 

88178 

88184 

88190 

762 

88195 

88  201 

88  207 

88  213 

88  218 

88  224 

88  230 

88  235 

88  241 

88  247 

763 

88  252 

88  258 

88  264 

88  270 

88  275 

88  281 

88  287 

88  292 

88  298 

88  304 

764 

88  309 

88  315 

88  321 

88  326 

88  332 

88  338 

88  343 

88  349 

88  355 

88  360 

765 

88  366 

88  372 

88  377 

88  383 

88  389 

88  395 

88  400 

88  406 

88  412 

88  417 

766 

88  423 

88  429 

88  434 

88  440 

88  446 

88  451 

88  457 

88  463 

88  468 

88  474 

767 

88  480 

88  485 

88  491 

88  497 

88  502 

88  508 

88  513 

88  519 

88  525 

88  530 

768 

88  536 

88  542 

88  547 

88  553 

88  559 

88  564 

88  570 

88  576 

88  581 

88  587 

769 

88  593 

88  598 

88  604 

88  610 

88  615 

88  621 

88  627 

88  632 

88  638 

88  643 

770 

88  649 

88  655 

88  660 

88  666 

88  672 

88  677 

88  683 

88  689 

88  694 

88  700 

771 

88  705 

88  711 

88  717 

88  722 

88  728 

88  734 

88  739 

88  745 

88  750 

88  756 

772 

88  762 

88  767 

88  773 

88  779 

88  784 

88  790 

88  795 

88  801 

88  807 

88  812 

773 

88  818 

88  824 

88  829 

88  835 

88  840 

88  846 

88  852 

88  857 

88  863 

88  868 

774 

88  874 

88  880 

88  885 

88  891 

88  897 

88  902 

88  908 

88  913 

88  919 

88  925 

775 

88  930 

88  936 

88  941 

88  947 

88  953 

88  958 

88  964 

88  969 

88  975 

88  981 

776 

88  986 

88  992 

88  997 

89  003 

89  009 

89  014 

89  020 

89  025 

89  031 

89  037 

777 

89  042 

89  048 

89  053 

89  059 

89  064 

89  070 

89  076 

89  081 

89  087 

89  092 

778 

89  098 

89104 

89109 

89115 

89120 

89126 

89131 

89137 

89143 

89148 

779 

89154 

89159 

89165 

89170 

89176 

89182 

89187 

89193 

89198 

89  204 

780 

89  209 

89  215 

89  221 

89  226 

89  232 

89  237 

89  243 

89  248 

89  254 

89  260 

781 

89  265 

89  271 

89  276 

89  282 

89  287 

89  293 

89  298 

89  304 

89  310 

89  315 

782 

89  321 

89  326 

89  332 

89  337 

89  343 

89  348 

89  354 

89  360 

89  365 

89  371 

783 

89  376 

89  382 

89  387 

89  393 

89  398 

89  404 

89  409 

89  415 

89  421 

89  426 

784 

89  432 

89  437 

89  443 

89  448 

89  454 

89  459 

89  465 

89  470 

89  476 

89  481 

785 

89  487 

89  492 

89  498 

89  504 

89  509 

89  515 

89  520 

89  526 

89  531 

89  537 

786 

89  542 

89  548 

89  553 

89  559 

89  564 

89  570 

89  575 

89  581 

89  586 

89  592 

787 

89  597 

89  603 

89  609 

89  614 

89  620 

89625 

89  631 

89  636 

89  642 

89  647 

788 

89  653 

89  658 

89  664 

89  669 

89  675 

89  680 

89  686 

89  691 

89  697 

89  702 

789 

89  708 

89  713 

89  719 

89  724 

89  730 

89  735 

89  741 

89  746 

89  752 

89  757 

790 

89  763 

89  768 

89  774 

89  779 

89  785 

89  790 

89  796 

89  801 

89  807 

89  812 

791 

89  818 

89  823 

89  829 

89  834 

89  840 

89  845 

89  851 

89  856 

89  862 

89  867 

792 

89  873 

89  878 

89  883 

89  889 

89  894 

89  900 

89  905 

89  911 

89  916 

89  922 

793 

89  927 

89  933 

89  938 

89  944 

89  949 

89  955 

89  960 

89  966 

89  971 

89  977 

794 

89  982 

89  988 

89  993 

89  998 

90  004 

90  009  90  015 

90  020 

90  026 

90  031 

795 

90  037 

90  042 

90  048 

90  053 

90  059 

90  064  90  069  90  075 

90  080 

90  086 

796 

90  091 

90  097 

90102 

90108 

90113 

90119 

90124 

90129 

90135 

90140 

797 

90146 

90  151 

90157 

90162 

90168 

90173 

90179 

90184 

90189 

90195 

798 

90  200 

90  206 

90  211 

90  217 

90  222 

90  227 

90  233 

90  238 

90  244 

90  249 

799 

90  255 

90  260 

90  266 

90  271 

90  276 

90  282 

90  287 

90  293 

90  298 

90  304 

800 

90  309 

90  314 

90  320 

90  325 

90  331 

90  336 

90  342 

90  347 

90  352 

90  358 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

760-800 


16 

800-860 

N 

O 

1 

2 

90  320 

3 

4 

5 

6 

90  342 

7 

8 

9 

90  358 

800 

90  309 

90  314 

90  325 

90  331 

90  336 

90  347 

90  352 

801 

90  363 

90  369 

90  374 

90  380 

90  385 

90  390 

90  396 

90  401 

90  407 

90  412 

802 

90  417 

90  423 

90  428 

90  434 

90  439 

90  445 

90  450 

90  455 

90  461 

90  466 

803 

90  472 

90  477 

90  482 

90  488 

90  493 

90  499 

90  504 

90  509 

90  515 

90  520 

■  804 

90  526 

90  531 

90  536 

90  542 

90  547 

90  553 

90  558 

90  563 

90  569 

90  574 

805 

90  580 

90  585 

90  590 

90  596 

90  601 

90  607 

90  612 

90  617 

90  623 

90  628 

806 

90  634 

90  639 

90  644 

90  650 

90  655 

90  660 

90  666 

90  671 

90  677 

90  682 

807 

90  687 

90  693 

90  698 

90  703 

90  709 

90  714 

90  720 

90  725 

90  730 

90  736 

808 

90  741 

90  747 

90  752 

90  757 

90  763 

90  768 

90  773 

90  779 

90  784 

90  789 

809 

90  79i 

90  800 

90  806 

90  811 

90  816 

90  822 

90  827 

90  832 

90  838 

90  843 

810 

90  849 

90  854 

90  859 

90  865 

90  870 

90  875 

90  881 

90  886 

90  891 

90  897 

811 

90  902 

90  907 

90  913 

90  918 

90  924 

90  929 

90  934 

90  940 

90  945 

90  950 

812 

90  956 

90  961 

90  966 

90  972 

90  977 

90  982 

90  988 

90  993 

90  998 

91004 

813 

91009 

91014 

91020 

91025 

91030 

91036 

91041 

91046 

91  052 

91057 

814 

91062 

91068 

91  073 

91078 

91084 

91089 

91094 

91100 

91105 

91110 

815 

91116 

91121 

91126 

91132 

91137 

91142 

91148 

91153 

91158 

91164 

816 

91169 

91174 

91180  91185 

91190 

91196 

91201 

91206 

91212 

91217 

817 

91222 

91228 

91233 

91238 

91243 

91249 

91254 

91259 

91265 

91270 

818 

91275 

91281 

91286 

91291 

91297 

91302 

91307 

91312 

91318 

91323 

819 

91328 

91334 

91339 

91344 

91350 

91355 

91360 

91365 

91371 

91376 

820 

91381 

91387 

91392 

91397 

91403 

91408 

91413 

91418 

91424 

91429 

821 

91434 

91440 

91445 

91450 

91455 

91461 

91466 

91471 

91477 

91482 

822 

91487 

91492 

91498 

91503 

91508 

91514 

91519 

91524 

91529  91535  | 

823 

91540 

91545 

91551 

91556 

91561 

91566 

91572 

91577 

91582 

91587 

824 

91593 

91598 

91603 

91609 

91614 

91619 

91624 

91630 

91635 

91640 

825 

91645 

91651 

91656 

91661 

91666 

91672 

91677 

91682 

91687 

91693 

826 

91698 

91703 

91709 

91714 

91719 

91724 

91730 

91735 

91740 

91745 

827 

91751 

91756 

91761 

91766 

91772 

91777 

91782 

91787 

91793 

91798 

828 

91803 

91808 

91814 

91819 

91824 

91829 

91834 

91840 

91845 

91850 

829 

91855 

91861 

91866 

91871 

91876 

91882 

91887 

91892 

91897 

91903 

830 

91908 

91913 

91918 

91924 

91929 

91934 

91939 

91944 

91950  91955  | 

831 

91960 

91965 

91971 

91976 

91981 

91986 

91991 

91997 

92  002 

92  007 

832 

92  012 

92  018 

92  023 

92  028 

92  033 

92  038 

92  044 

92  049 

92  054 

92  059 

833 

92  065 

92  070 

92  075 

92  080 

92  085 

92  091 

92  096 

92  101 

92  106 

92111 

834 

92  117 

92  122 

92127 

92132 

92137 

92143 

92  148 

92  153 

92  158 

92163 

835 

92169 

92  174 

92179 

92  184 

92  189 

92  195 

92 '200 

92  205 

92  210 

92  215 

836 

92  221 

92  226 

92  231 

92  236 

92  241 

92  247 

92  252 

92  257 

92  262 

92  267 

837 

92  273 

92  278 

92  283 

92  288 

92  293 

92  298 

92  304 

92  309 

92  314 

92  319 

838 

92  324 

92  330 

92  335 

92  340 

92  345 

92  350 

92  355 

92  361 

92  366 

92  371 

839 

92  376 

92  381 

92  387 

92  392 

92  397 

92  402 

92  407 

92  412 

92  418 

92  423 

840 

92  428 

92  433 

92  438 

92  443 

92  449 

92  454 

92  459 

92  464 

92  469 

92  474 

841 

92  480 

92  485 

92  490 

92  495 

92  500 

92  505 

92  511 

92  516 

92  521 

92  526 

842 

92  531 

92  536 

92  542 

92  547 

92  552 

92  557 

92  562 

92  567 

92  572 

92  578 

843 

92  583 

92  588 

92  593 

92  598 

92  603 

92  609 

92  614 

92  619 

92  624 

92  629 

844 

92  634 

92  639 

92  645 

92  650 

92  655 

92  660 

92  665 

92  670 

92  675 

92  681 

845 

92  686 

92  691 

92  696 

92  701 

92  706 

92  711 

92  716 

92  722 

92  727 

92  732 

846 

92  737 

92  742 

92  747 

92  752 

92  758 

92  763 

92  768 

92  773 

92  778 

92  783 

847 

92  788 

92  793 

92  799 

92  804 

92  809 

92  814 

92  819 

92  824 

92  829 

92  834 

848 

92  840 

92  845 

92  850  92  855 

92  860 

92  865 

92  870 

92  875 

92  881 

92  886 

849 

92  891 

92  896 

92  901 

92  906 

92  911 

92  916 

92  921 

92  927 

92  932 

92  937 

850 

92  942 

92  947 

92952 

92  957 

92  962 

92  967 

92  973 

92  978 
7 

92  983 

92  988 

N 

O 

1 

2 

3 

4 

5 

6 

8 

9 

800-860 


860-900 

17 

N 

O 

1 

2 
92  952 

3 

4 

5 

6 

7 

8 

9 

850 

92  942 

92  947 

92  957 

92  962 

92  967 

92  973 

92  978 

92  983 

92  988 

851 

92  993 

92  998 

93  003 

93  008 

93  013 

93  018 

93  024 

93  029 

93  034 

93  039 

852 

93  044 

93  049 

93  054 

93  059 

93  064 

93  069 

93  075 

93  080 

93  085 

93  090 

853 

93  095 

93  100 

93  105 

93  110 

93  115 

93  120 

93  125 

93  131 

93  136 

93  141 

854 

93  146 

93  151 

93  156 

93  161 

93  166 

93  171 

93  176 

93  181 

93  186 

93  192 

855 

93  197 

93  202 

93  207 

93  212 

93  217 

93  222 

93  227 

93  232 

93  237 

93  242 

856 

93  247 

93  252 

93  258 

93  263 

93  268 

93  273 

93  278 

93  283 

93  288 

93  293 

857 

93  298 

93  303 

93  308 

93  313 

93  318 

93  323 

93  328 

93  334 

93  339 

93  344 

858 

93  349 

93  354 

93  359 

93  364 

93  369 

93  374 

93  379 

93  384 

93  389 

93  394 

859 

93  399 

93  404 

93  409 

93  414 

93  420 

93  425 

93  430 

93  435 

93  440 

93  445 

860 

93  450 

93  455 

93  460  93  465 

93  470 

93  475 

93  480 

93  485 

93  490 

93  495 

861 

93  500 

93  505 

93  510 

93  515 

93  520 

93  526 

93  531 

93  536 

93  541 

93  546 

862 

93  551 

93  556 

93  561 

93  566 

93  571 

93  576 

93  581 

93  586 

93  591 

93  596 

863 

93  601 

93  606 

93  611 

93  616 

93  621 

93  626 

93  631 

93  636 

93  641 

93  646 

864 

93  651 

93  656 

93  661 

93  666 

93  671 

93  676 

93  682 

93  687 

93  692 

93  697 

865 

93  702 

93  707 

93  712 

93  717 

93  722 

93  727 

93  732 

93  737 

93  742 

93  747 

866 

93  752 

93  757 

93  762 

93  767 

93  772 

93  777 

93  782 

93  787 

93  792 

93  797 

867 

93  802 

93  807 

93  812 

93  817 

93  822 

93  827 

93  832 

93  837 

93  842 

93  847 

868 

93  852 

93  857 

93  862 

93  867 

93  872 

93  877 

93  882 

93  887 

93  892 

93  897 

869 

93  902 

93  907 

93  912 

93  917 

93  922 

93  927 

93  932 

93  937 

93  942 

93  947 

870 

93  952 

93  957 

93  962 

93  967 

93  972 

93  977 

93  982 

93  987 

93  992 

93  997 

871 

94  002 

94  007 

94  012 

94  017 

94  022 

94  027 

94  032 

94  037 

94  042 

94  047 

872 

94  052 

94  057 

94  062 

94  067 

94  072 

94  077 

94  082 

94  086 

94  091 

94  096 

873 

94  101 

94  106 

94  111 

94  116 

94121 

94  126 

94131 

94  136 

94  141 

94  146 

874 

94  151 

94156 

94  161 

94  166 

94  171 

94176 

94181 

94  186 

94  191 

94196 

875 

94  201 

94  206 

94  211 

94  216 

94  221 

94  226 

94  231 

94  236 

94  240 

94  245 

876 

94  250 

94  255 

94  260 

94  265 

94  270 

94  275 

94  280 

94  285 

94  290 

94  295 

877 

94  300  94  305 

94  310 

94  315 

94  320 

94  325 

94  330 

94  335 

94  340  94  345  | 

878 

94  349 

94  354 

94  359 

94  364 

94  369 

94  374 

94  379 

94  384 

94  389 

94  394 

879 

94  399 

94  404 

94  409 

94  414 

94  419 

94  424 

94  429 

94  433 

94  438 

94  443 

880 

94  448 

94  453 

94  458 

94  463 

94  468 

94  473 

94  478 

94  483 

94  488 

94  493 

881 

94  498 

94  503 

94  507 

94  512 

94  517 

94  522 

94  527 

94  532 

94  537 

94  542 

882 

94  547 

94  552 

94  557 

94  562 

94  567 

94  571 

94  576 

94  581 

94  586 

94  591 

883 

94  596 

94  601 

94  606 

94  611 

94  616 

94  621 

94  626 

94  630 

94  635 

94  640 

884 

94  645 

94  650 

94  655 

94  660 

94  665 

94  670  94  675 

94  680 

94  685 

94  689 

885 

94  694 

94  699 

94  704 

94  709 

94  714 

94  719 

94  724 

94  729 

94  734 

94  738 

886 

94  743 

94  748 

94  753 

94  758 

94  763 

94  768 

94  773 

94  778 

94  783 

94  787 

887 

94  792 

94  797 

94  802 

94  807 

94  812 

94  817 

94  822 

94  827 

94  832 

94  836 

888 

94  841 

94  846 

94  851 

94  856 

94  861 

94  866 

94  871 

94  876 

94  880 

94  885 

889 

94  890 

94  895 

94  900 

94  905 

94  910 

94  915 

94  919 

94  924 

94  929 

94  934 

890 

94  939 

94  944 

94  949 

94  954 

94  959 

94  963 

94  968 

.  94  973 

94  978 

94  983 

891 

94  988 

94  993 

94  998 

95  002 

95  007 

95  012 

95017 

95  022 

95  027 

95  032 

892 

95  036 

95  041 

95  046 

95  051 

95  056 

95  061 

95  066 

95  071 

95  075 

95  080 

893 

95  085 

95  090 

95  095 

95  100 

95  105 

95  109 

95  114 

95  119 

95  124 

95  129 

894 

95  134 

95  139 

95  143 

95  148 

95  153 

95  158 

95  163 

95  168 

95  173 

95  177 

895 

95  182 

95  187 

95  192 

95  197 

95  202 

95  207 

95  211 

95  216 

95  221 

95  226 

896 

95  231 

95  236 

95  240 

95  245 

95  250 

95  255 

95  260 

95  265 

95  270 

95  274 

897 

95  279 

95  284 

95  289 

95  294 

95  299 

95  303 

95  308 

95  313 

95  318 

95  323 

898 

95  328 

95  332 

95  337 

95  342 

95  347 

95  352 

95  357 

95  361 

95  366 

95  371 

899 

95  376 

95  381 

95  386 

95  390 

95  395 

95  400 

95  405 

95  410 

95  41i 

95  419 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

860-900 


18 


900-960 


:n- 

0 

1    2 

3 

4 

5 

6 

7 

8 

9 

900 

95  424 

95  429  95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

901 

95  472 

95  477  95  482 

95  487 

95  492 

95  497 

95  501 

95  506 

95  511 

95  516 

902 

95  521 

95  525  95  530 

95  535 

95  540 

95  545 

95  550 

95  554 

95  559 

95  564 

903 

95  569 

95  574  95  578 

95  583 

95  588 

95  593 

95  598 

95  602 

95  607 

95  612 

904 

95  617 

95  622  95  626 

95  631 

95  636 

95  641 

95  646 

95  650 

95  655 

95  660 

905 

95  665 

95  670  95  674 

95  679 

95  684 

95  689 

95  694 

95  698 

95  703 

95  708 

906 

95  713 

95  718  95  722 

95  727 

95  732 

95  737 

95  742 

95  746 

95  751 

95  756 

907 

95  761 

95  766  95  770 

95  775 

95  780 

95  785 

95  789 

95  794 

95  799 

95  804 

908 

95  809 

95  813  95  818 

95  823 

95  828 

95  832 

95  837 

95  842 

95  847 

95  852 

909 

95  856 

95  861  95  866 

95  871 

95  875 

95  880 

95  885 

95  890 

95  895 

95  899 

910 

95  904 

95  909  95  914 

95  918 

95  923 

95  928 

95  933 

95  938 

95  942 

95  947 

911 

95  952 

95  957  95  961 

95  966 

95  971 

95  976 

95  980 

95  985 

95  990 

95  995 

912 

95  999 

96  004  96  009 

96  014 

96  019 

96  023 

96  028 

96  033 

96  038 

96  042 

913 

96  047 

96  052  96  057 

96  061 

96  066 

96  071 

96  076 

96  080 

96  085 

96  090 

914 

96  095 

96  099  96104 

96109 

96114 

96118 

96123 

96128 

96133 

96137 

915 

96142 

96147  96]  52 

96156 

96161 

96166 

96  171 

96175 

96180 

96185 

916 

96190 

96194  96199 

96  204 

96  209 

96  213 

96  218 

96  223 

96  227 

96  232 

917 

96  237 

96  242  96  246 

96  251 

96  256 

96  261 

96  265 

96  270 

96  275 

96  280 

918 

96  284 

96  289  96  294 

96  298 

96  303 

96  308 

96  313 

96  317 

96  322 

96  327 

919 

96  332 

96  336  96  341 

96  346 

96  350 

96  355 

96  360 

96  365 

96  369 

96  374 

920 

96  379 

96  384  96  388 

96  393 

96  398 

96  402 

96  407 

96  412 

96  417 

96  421 

921 

96  426 

96  431  96  435 

96  440 

96  445 

96  450 

96  454 

96  459 

96  464 

96  468 

922 

96  473 

96  478  96  483 

96  487 

96  492 

96  497 

96  501 

96  506 

96  511 

96  515 

923 

96  520 

96  525  96  530 

96  534 

96  539 

96  544 

96  548 

96  553 

96  558 

96  562 

924 

96  567 

96  572  96  577 

96  581 

96  586 

96  591 

96  595 

96  600 

96  605 

96  609 

925 

96  614 

96  619  96  624 

96  628 

96  633 

96  638 

96  642 

96  647 

96  652 

96  656 

926 

96  661 

96  666  96  670 

96  675 

96  680 

96  685 

96  689 

96  694 

96  699 

96  703 

927 

96  708 

96  713  96  717 

96  722 

96  727 

96  731 

96  736 

96  741 

96  745 

96  750 

928 

96  755 

96  759  96  764 

96  769 

96  774 

96  778 

96  783 

96  788 

96  792 

96  797 

929 

96  802 

96  806  96  811 

96  816 

96  820 

96  825 

96  830 

96  834 

96  839 

96  844 

930 

96  848 

96  853  96  858 

96  862 

96  867 

96  872 

96  876 

96  881 

96  886 

96  890 

931 

96  895 

96  900  96  904 

96  909 

96  914 

96  918 

96  923 

96  928 

96  932 

96  937 

932 

96  942 

96  946  96  951 

96  956 

96  960 

96  965 

96  970 

96  974 

96  979 

96  984 

933 

96  988 

96  993  96  997 

97  002 

97  007 

97  011 

97  016 

97  021 

97  025 

97  030 

934 

97  035 

97  039  97  044 

97  049 

97  053 

97  058 

97  063 

97  067 

97  072 

97  077 

935 

97  081 

97  086  97  090 

97  095 

97100 

97104 

97109 

97114 

97118 

97  123 

936 

97128 

97132  97137 

97142 

97  146 

97151 

97155 

97  160 

97  165 

97169 

937 

97174 

97179  97183 

97188 

97  192 

97197 

97  202 

97  206 

97  211 

97  216 

938 

97  220 

97  225  97  230 

97  234 

97  239 

97  243 

97  248 

97  253 

97  257 

97  262 

939 

97  267 

97  271  97  276 

97  280 

97  285 

97  290 

97  294 

97  299 

97  304 

97  308 

940 

97  313 

97  317  97  322 

97  327 

97  331 

97  336 

97  340 

97  345 

97  350 

97  354 

941 

97  359 

97  364  97  368 

97  373 

97  377 

97  382 

97  387 

97  391 

97  396 

97  400 

942 

97  405 

97  410  97  414 

97  419 

97  424 

97  428 

97  433 

97  437 

97  442 

97  447 

943 

97  451 

97  456  97  460 

97  465 

97  470 

97  474 

97  479 

97  483 

97  488 

97  493 

944 

97  497 

97  502  97  506 

97  511 

97  516 

97  520 

97  525 

97  529 

97  534 

97  539 

945 

97  543 

97  548  97  552 

97  557 

97  562 

97  566 

97  571 

97  575 

97  580 

97  585 

946 

97  589 

97  594  97  598 

97  603 

97  607 

97  612 

97  617 

97  621 

97  626 

97  630 

947 

97  635 

97  640  97  644 

97  649 

97  653 

97  658 

97  663 

97  667 

97  672 

97  676 

948 

97  681 

97  685  97  690 

97  695 

97  699 

97  704 

97  708 

97  713 

97  717 

97  722 

949 

97  727 

97  731  97  736 

97  740 

97  745 

97  749 

97  754 

97  759 

97  763 

97  768 

950 

97  772 

97  777  97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

N 

0 

1    2 

3 

4 

5 

6 

7 

8 

9 

900-960 


950-1000 

19 

N 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

951 

97  818 

97  823 

97  827 

97  832 

97  836 

97  841 

97  845 

97  850 

97  855 

97  859 

952 

97  864 

97  868 

97  873 

97  877 

97  882 

97  886 

97  891 

97  896 

97  900 

97  905 

953 

97  909 

97  914 

97  918 

97  923 

97  928 

97  932 

97  937 

97  941 

97  946 

97  950 

954 

97  955 

97  959 

97  964 

97  968 

97  973 

97  978 

97  982 

97  987 

97  991 

97  996 

955 

98  000 

98  005 

98  009 

98  014 

98  019 

98  023 

98  028 

98  032 

98  037 

98  041 

956 

98  046 

98  050 

98  055 

98  059 

98  064 

98  068 

98  073 

98  078 

98  082 

98  087 

957 

98  091 

98  096  98100  98105 

98109 

98114 

98118 

98123 

98127 

98132 

958 

98137 

98141 

98146 

98150 

98155 

98159 

98164 

98168 

98173 

98177 

959 

98  182 

98186 

98191 

98195 

98  200 

98  204 

98  209 

98  214 

98  218 

98  223 

960 

98  227 

98  232 

98  236 

98  241 

98  245 

98  250 

98  254 

98  259 

98  263 

98  268 

961 

98  272 

98  277 

98  281 

98  286 

98  290 

98  295 

98  299 

98  304 

98  308 

98  313 

962 

98  318 

98  322 

98  327 

98  331 

98  336 

98  340 

98  345 

98  349 

98  354 

98  358 

963 

98  363 

98  367 

98  372 

98  376 

98  381 

98  385 

98  390 

98  394 

98  399 

98  403 

964 

98  408 

98  412 

98  417 

98  421 

98  426 

98  430  98  435 

98  439 

98  444 

98  448 

965 

98  453 

98  457 

98  462 

98  466 

98  471 

98  475 

98  480 

98  484 

98  489 

98  493 

966 

98  498 

98  502 

98  507 

98  511 

98  516 

98  520 

98  525 

98  529 

98  534 

98  538 

967 

98  543 

98  547 

98  552 

98  556 

98  561 

98  565 

98  570 

98  574 

98  579 

98  583 

968 

98  588 

98  592 

98  597 

98  601 

98  605 

98  610 

98  614 

98  619 

98  623 

98  628 

969 

98  632 

98  637 

98  641 

98  646 

98  650 

98  655 

98  659 

98  664 

98  668 

98  673 

970 

98  677 

98  682 

98  686 

98  691 

98  695 

98  700 

98  704 

98  709 

98  713 

98  717 

971 

98  722 

98  726 

98  731 

98  735 

98  740 

98  744 

98  749 

98  753 

98  758 

98  762 

972 

98  767 

98  771 

98  776 

98  780 

98  784 

98  789 

98  793 

98  798 

98  802 

98  807 

973 

98  811 

98  816 

98  820 

98  825 

98  829 

98  834 

98  838 

98  843 

98  847 

98  851 

974 

98  856  98  860  98  865 

98  869 

98  874 

98  878 

98  883 

98  887 

98  892 

98  896 

975 

98  900 

98  905 

98  909 

98  914 

98  918 

98  923 

98  927 

98  932 

98  936 

98  941 

976 

98  945 

98  949 

98  954 

98  958 

98  963 

98  967 

98  972 

98  976 

98  981 

98  985 

977 

98  989 

98  994 

98  998 

99  003 

99  007 

99  012 

99  016 

99  021 

99  025 

99  029 

978 

99  034 

99  038 

99  043 

99  047 

99  052 

99  056 

99  061 

99  065 

99  069 

99  074 

979 

99  078 

99  083 

99  087 

99  092 

99  096 

99100  99105 

99109 

99114 

99118 

980 

99123 

99127 

99131 

99136 

99140 

99145 

99149 

99154 

99158 

99162 

981 

99167 

99171 

99176 

99  180 

99  185 

99189 

99193 

99198 

99  202 

99  207 

982 

99  211 

99  216 

99  220 

99  224 

99  229 

99  233 

99  238 

99  242 

99  247 

99  251 

983 

99  255 

99  260 

99  264 

99  269 

99  273 

99  277 

99  282 

99  286 

99  291 

99  295 

984 

99  300 

99  304 

99  308 

99  313 

99  317 

99  322 

99  326 

99  330 

99  335 

99  339 

985 

99  344 

99  348 

99  352 

99  357 

99  361 

99  366 

99  370 

99  374 

99  379 

99  383 

986 

99  388 

99  392 

99  396 

99  401 

99  405 

99  410 

99  414 

99  419 

99  423 

99  427 

987 

99  432 

99  436 

99  441 

99  445 

99  449 

99  454 

99  458 

99  463 

99  467 

99  471 

988 

99  476 

99  480 

99  484 

99  489 

99  493 

99  498 

99  502 

99  506 

99  511 

99  515 

989 

99  520 

99  524 

99  528 

99  533 

99  537 

99  542 

99  546  99  550  99  555 

99  559 

990 

99  564 

99  568 

99  572 

99  577 

99  581 

99  585 

99  590 

99  594 

99  599 

99  603 

991 

99  607 

99  612 

99  616 

99  621 

99  625 

99  629 

99  634 

99  638 

99  642 

99  647 

992 

99  651 

99  656 

99  660 

99  664 

99  669 

99  673 

99  677 

99  682 

99  686 

99  691 

993 

99  695 

99  699 

99  704 

99  708 

99  712 

99  717 

99  721 

99  726 

99  730 

99  734 

994 

99  739 

99  743 

99  747 

99  752 

99  756 

99  760 

99  765 

99  769 

99  774 

99  778 

995 

99  782 

99  787 

99  791 

99  795 

9^800 

99  804 

99  808 

99  813 

99  817 

99  822 

996 

99  826 

99  830 

99  835 

99  839 

99  843 

99  848 

99  852 

99  856 

99  861 

99  865 

997 

99  870 

99  874 

99  878 

99  883 

99  887 

99  891 

99  896 

99  900 

99  904 

99  909 

998 

99  913 

99  917 

99  922 

99  926 

99  930 

99  935 

99  939 

99  944 

99  948 

99  952 

999 

99  957 

99  961 

99  965 

99  970 

99  974 

99  978 

99  983 

99  987 

99  991 

99  996 

1000 

00  000 

00  004 

00  009 

00  013 

00  017 
4 

00  022 

00  026 

00  030 

00  03i 

00039 
9 

N 

o 

1 

2 

3 

5 

6 

7 

8 

950-1000 


20     TABLE  IL-LOGAEITHMS  OF  CONSTANTS. 

Circomference  of  the  Circle  in  degrees —           360 

log 
2.55  630  250    " 
4.33  445  375 
6.11260  500 

0.49  714  9S7 

Circumference  of  the  Circle  in  minutes      —      21  600 

Circumference  of  the  Circle  in  seconds =  1  296  000 

If  the  radius  r  =  1,  half  the  Circumference  of  the  Circle  is 
X  =  3. 14  159  265  358  979  323  846  26+  338  3ZS 

Also: 
2ir=   6.28318531 

log 
0.79817  987 

x2  =  9.  86960440 

log 
0.99429975 

4x=  12.56637  061 
^-    1.57  079633 

1.09920986 
0.19611988 

1  =  0.10132118 

9.00570025-10 

1=    1.M719  755 
3 

^=   4.18  879020 
3 

1=   0.78  539816 

0. 02  002  862 

V*-- 1.77  245  385 
—  =  0.56418958 

0.24  857494 

9.  75  142  506  -  10 

0.62  208  861 

V' 

9. 89  508  988-10 

1=0.97  720502 

9.98998  569-10 

1=   0.52359878 
o 

9.71899862-10 

^^  =  1.12  837  917 

0. 05  245  506 

1=   0.31830989 

9.50285013-10 

^x  =  1.46459189 

0.16571662 

^=   0.15  915  494 

9.20182013-10 

-1=0.68  278406 

9.83428338-  10 

-=   0.95492966 

9.97997138-10 

^t2  =  2. 14  502  940 

0.33  143  32i 

^=    1.27323954 

0.10491012 

i/'^  =  0.62  035  049 

9.  79  263  713  -  10 

^=   0.23873241 

9.37  791139-10 

^1  =  0.80599  598 

9.90633  287-10 

Aic  a,  whose  length  is  equal  to  the  radius  r,  is  : 

log 

in  degrees aP =  1^ =  57.29577951°. 

1. 75  812  263 

in  minutes a' _  10  800 -  3  437.  74  677'   . 

3.53  627  388 

in  seconds a" _  648  000 -  2O6  264. 806"  . . 

X 

5.31442  513 

Arc  2a,  whose  length  is  equal  to  twrice  the  radius,  2r,  is  : 

in  degrees 2a9  ....  =  ^ =  114.59155  903° 

2.05  915  263 

in  minutes 2  a' 21 600           —  6S75  49'??4' 

3.83  730388 
5.61545  513 

w 
in  seconds 2a" . . . .  -  ^  ^^^^  . . .  =  412  529. 612" . . 

If  the  radius  r  =  1,  the  length  of  the  arc  is : 

for  1  degree \ -   ''   -  0. 01  745  329. . . 

8.  24  1S7  737  -  10 

for  1  minute . . . 

a"               180 
..^ —      ^                 — 000  029  089 

6. 46  372  612  -  10 

a'                10800 

for  1  second 

1                                        V 

....-0.00000485... 

4. 68  557  487 -10 

a" 648000' 

for  J  degree ^■■'=^ =  0. 00 872 66i. . . 

7.  94084  737-10 

for  1  minute. . . 

..        —      '                      000  014  ^44 

6. 16  269  612  -  10 

"2a' 2Um U.WUlloll... 

for  J  second . . . 

1                                          X 

....  =  0.00000  242... 

4. 38  454  487  -  10 

2a"             1296000 

Sin  1"  in  the  unit  cL 

role -0  00  000  485 

4.  68  557  487  -  10 

21 


1 

TA 
'HE  L( 

ELE  r 

[I 

[THMS 

3GAE] 

or  TBDB 

trigoxo:metkic 

FU^CTIOXS: 

Prom  O'  to  0°  3',  or  89^  57'  to  90^,  for  every  second , 

Prom  0°  to  2-,  or  88*^  to  90°,  foi 

every  ten  seconds; 

Prom  1=  U 
Notk.    T 

log:  sin 

a  89^,  for  every  minute 

3. 

0  is  to  be  appended. 

]i«eM  =  10.00000 

o  all  the  logarithms  -1 

0° 

o 

0' 

1' 

O  r 

t  f 

f  f 

O'              1' 

o  * 

ft 



6. 46  373 

6.76476 

60 

30 

6.16270   6.63982 

6  86167 

30 

1 

4.68  557 

6-47090 

6.76  836 

59 

31 

6.1769f   6.64462 

6  86455 

29 

2 

4.98660 

6.47  797 

6.  77  193 

58 

32 

6.19072    6.64936 

6  86  742 

28 

3 

5. 16  270 

6.48492 

6.77  548 

57 

ZZ 

6.20409    6.65406 

6  87(^7 

27 

4 

5.28  763 

6.49175 

6.77900 

56 

34 

6.21705    6.65  870 

6  87310 

26 

o 

5.38454 

6.49  849 

6.78248 

55 

35 

6.2296f   6.66330 

6. 87  591 

25 

6 

5.46373 

6.50512 

6.7859i 

54 

36 

6.24188   6.66785 

6  87870 

24 

? 

5. 53  067 

6.51165 

6.78938 

53 

37 

6.25  378   6.67235 

688147 

23 

S 

5.58866 

6.51808 

6.79278 

52 

38 

6.26536   6.67680 

6.88423 

22 

9 

5.63  982 

6.52  442 

6.79616 

51 

39 

6.27664   6.68121 

688697 

21 

lO 

5. 68  557 

6. 53  067 

6.79952 

50 

40 

6.28763    6.68557 

688969 

20 

11 

5.72  697 

6.53  6S3 

6.80285 

49 

41 

6.29836   6.68990 

689240 

19 

12 

5.76476 

6.  54  291 

6.80615 

48 

42 

6.30882   6.69418 

689509 

18 

13 

5.79952 

6.54  890 

6.80943 

47 

43 

6.3190f    6.69841 

6.89  776 

17 

14 

5. 83  170 

6.  55  481 

6.81268 

46 

44 

6.32903    6.70261 

6  90042 

16 

15 

5.86167 

6.56064 

6-81591 

45 

45 

6.33879   6.70676 

690306 

15 

16 

5.88969 

6.56639 

6.81911 

44 

46 

6.34833    6.710SS 

690568 

14 

17 

5.91602 

6.57  207 

6.82230 

43 

47 

6.35767   6.71496 

690829 

13 

18 

5.94085 

6-57  767 

6.82545 

42 

48 

6.36682   6  71900 

691068 

12 

19 

5.%  433 

6-58320 

6-82859 

41 

49 

6.37577    6  72300 

691346 

11 

20 

5.98  660 

6.58  866 

6.83170 

40 

50 

638454   6.72697 

691602 

lO 

— _ 

6. 00  779 

6.59406 

6.83  479 

39 

51 

6.39315    6  73090 

6  91857 

9 

:: 

6.02800 

6.59939 

6.83  786 

38 

52 

640158   673479 

6  92110 

8 

23 

6.04  730 

6-60465 

6.84091 

37 

53 

6.40985    6  73  865 

692362 

7 

24 

6.06  579 

6.60985 

6.84  394 

36 

54 

641797   6  74248 

692612 

6 

25 

6.  OS  351 

6-61499 

6.84694 

35 

55 

642594    674627 

692861 

5 

26 

6.10055 

6.62  007 

6.84993 

34 

56 

643376   6.75003 

693109 

4 

27 

6.1169f 

6.62  509 

6.85  289 

33 

57 

6  44145    6  75  376 

693  355 

3 

28 

6.13  273 

6.63  006 

6.85  584 

32 

58 

644900   6  75  746 

693  599 

2 

29 

6.14  797 

6.63  496 

6. 85  876 

31 

59 

64S643    6  76112 

693  843 

1 

30 

6. 16  270 

6.63982 

6-86167 

30 

eo 

6.46373    6.76476 

694065 

O 

59' 

58' 

57' 

»» 

f  f 

59'          58' 

57' 

ff 

lo|^oofc  =  ]ogcos 
kg  sia^  10. 00  000 


89 


log  cos 


22 

0° 

t  ff 

log  sin 

log  tan 

log  COS 

ff    f 

f  ff 

log  sin 

log  tan 

log  cos 

ff  f 

O  0 





10.00000 

0  6O 

lOO 

7.46  373 

7.46  373 

10.00000 

0  5O 

10 

5.  68  557 

5.  68  557 

10.00000 

50 

10 

7.  47  090 

7. 47  091 

10.00000 

50 

20 

5.98  660 

5.  98  660 

10.00000 

40 

20 

7. 47  797 

7.  47  797 

10.00000 

40 

30 

6. 16  270 

6. 16  270 

10.00000 

30 

30 

7.  48  491 

7.  48  492 

10.00000 

30 

40 

6.  28  763 

6.  28  763 

10.00000 

20 

40 

7.49175 

7.  49  176 

10.00000 

20 

50 

6.  38  454 

6.  38  454 

10.00000 

10 

50 

7, 49  849 

7.49  849 

10.00000 

10 

1  0 

6.  46  373 

6.  46  373 

10.00000 

0  59 

110 

7.50  512 

7.50  512 

10.00000 

0  49 

10 

6.  53  067 

6.  53  067 

10.00000 

50 

10 

7.  51 165 

7.  51 165 

10.00000 

50 

20 

6.  58  866 

6.  58  866 

10.00000 

40 

20 

7.  51  808 

7.51809 

10.00000 

40 

30 

6.  63  982 

6.  63  982 

10.00000 

30 

30 

7.  52  442 

7.52  443 

10.00000 

30 

40 

6.  68  557 

6.68  557 

10.00000 

20 

40 

7.  53  067 

7.  53  067 

10.00000 

20 

50 

6.  72  697 

6.  72  697 

10.00000 

10 

50 

7.  53  683 

7.  53  683 

10.00000 

10 

2  0 

6.76  476 

6.76  476 

10.00000 

0  58 

12  0 

7.  54  291 

7.  54  291 

10.00000 

048 

10 

6.  79  952 

6.  79  952 

10.00000 

50 

10 

7.  54  890 

7.  54  890 

10.00000 

50 

20 

6.  83  170 

6.  83  170 

10.00000 

40 

20 

7.  55  481 

7.  55  481 

10.00000 

40 

30 

6.  86  167 

6.  86  167 

10,00000 

30 

30 

7.  56  064 

7.56064 

10.00000 

30 

40 

6.  88  969 

6.  88  969 

10.00000 

20 

40 

7.  56  639 

7.  56  639 

10.00000 

20 

50 

6.  91  602 

6.  91  602 

10.00000 

10 

50 

7.  57  206 

7.  57  207 

10.00000 

10 

3  0 

6.  94  085 

6.  94  085 

10.00000 

057 

13  0 

7.  57  767 

7.  57  767 

10.00000 

047 

10 

6.  96  433 

6.96  433 

10.00000 

50 

10 

7.  58  320 

7.  58  320 

10.00000 

50 

20 

6.98  660 

6.  98  661 

10.00000 

40 

20 

7.  58  866 

7.  58  867 

10.00000 

40 

30 

7. 00  779 

7.  00  779 

10.00000 

30 

30 

7.59  406 

7.59  406 

10.00000 

30 

40 

7.  02  800 

7.02  800 

10.00000 

20 

40 

7.59  939 

7.59  939 

10.00000 

20 

50 

7.04  730 

7.  04  730 

10.00000 

10 

50 

7.  60  465 

7.  60  466 

10.00000 

10 

4  0 

7.  06  579 

7.06  579 

10.00000 

0  56 

14  0 

7.60  985 

7.  60  986 

10.00000 

0  46 

10 

7.08  351 

7.  08  352 

10.00000 

50 

10 

7.  61  499 

7.  61  500 

10.00000 

50 

20 

7.10  055 

7.10  055 

10.00000 

40 

20 

7.  62  007 

7.  62  008 

10.00000 

40 

30 

7. 11  694 

7.11694 

10.00000 

30 

30 

7.  62  509 

7.62  510 

10.00000 

30 

40 

7. 13  273 

7. 13  273 

10.00000 

20 

40 

7.  63  006 

7.  63  006 

10.00000 

20 

50 

7. 14  797 

7. 14  797 

10.00000 

10 

50 

7.  63  496 

7.  63  497 

10.00000 

10 

5  0 

7.16  270 

7.16  270 

10.00000 

0  55 

15  0 

7.  63  982 

7.  63  982 

10.00000 

0  45 

10 

7. 17  694 

7. 17  694 

10.00000 

50 

10 

7.  64  461 

7.  64  462 

10.00000 

50 

20 

7. 19  072 

7.19  073 

10.00000 

40 

20 

7.  64  936 

7.  64  937 

10.00000 

40» 

30 

7.20  409 

7.20  409 

10.00000 

30 

30 

7.65  406 

7.  65  406 

10.00000 

30 

40 

7.  21  705 

7.  21  705 

10.00000 

20 

40 

7.65  870 

7.  65  871 

10.00000 

20 

50 

7.  22  964 

7.  22  964 

10.00000 

10 

50 

7. 66  330 

7.  66  330 

10.00000 

10 

6  0 

7^4  188 

7.  24  188 

10.00000 

0  54 

16  0 

7.  66  784 

7.  66  785 

10.00000 

0  44 

10 

7.  25  378 

7.  25  378 

10.00000 

50 

10 

7.  67  235 

7.  67  235 

10.00000 

50 

20 

7.  26  536 

7.26536 

10.00000 

40 

20 

7.  67  680 

7.67  680 

10.00000 

40 

30 

7.  27  664 

7.  27  664 

10.00000 

30 

30 

7.  68  121 

7.68121 

10.00000 

30 

40 

7.  28  763 

7.  28  764 

10.00000 

20 

40 

7.68  557 

7.  68  558 

9.99999 

20 

50 

7.  29  836 

7.29  836 

10.00000 

10 

50 

7.68  989 

7.68  990 

9.99999 

10 

7  0 

7.  30  882 

7.30  882 

10.00000 

0  53 

170 

7.69  417 

7.69  418 

9.99  999 

0  43 

10 

7.  31  904 

7.  31  904 

10.00000 

50 

10 

7.  69  841 

7.69  842 

9.  99  999 

50 

20 

7.32  903 

7. 32  903 

10.00000 

40 

20 

7.  70  261 

7.  70  261 

9. 99  999 

40 

30 

7.  33  879 

7. 33  879 

10.00000 

30 

30 

7.  70  676 

7.  70  677 

9.  99  999 

30 

40 

7.  34  833 

7.34  833 

10.00000 

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45 

16 

83  594 

97  371 

02  629 

86  223 

44 

17 

83  608 

97  396 

02  604 

86  211 

43 

18 

83  621 

97  421 

02  579 

86  200 

42 

19 

83  634 

97  447 

02  553 

86188 

41 

20 

83  648 

97  472 

02  528 

86176 

40 

21 

83  661 

97  497 

02  503 

86164 

39 

22 

83  674 

97  523 

02  477 

86152 

38 

23 

83  688 

97  548 

02  452 

86140 

37 

24 

83  701 

97  573 

02  427 

86128 

36 

25 

83  715 

97  598 

02  402 

86116 

35 

26 

83  728 

97  624 

02  376 

86104 

34 

27 

83  741 

97  649 

02  351 

86  092 

ZZ 

28 

83  755 

97  674 

02  326 

86  080 

32 

29 

83  768 

97  700 

02  300 

86  068 

31 

30 

83  781 

97  72i 

02  275 

86  056 

30 

31 

83  795 

97  750 

02  250 

86  044 

29 

32 

83  808 

97  776 

02  224 

86  032 

28 

33 

83  821 

97  801 

02  199 

86  020 

27 

34 

83  834 

97  826 

02  174 

86  008 

26 

35 

83  848 

97  851 

02  149 

85  996 

25 

36 

83  861 

97  877 

02123 

85  984 

24 

37 

83  874 

97  902 

02  098 

85  972 

23 

38 

83  887 

97  927 

02  073 

85  960 

22 

39 

83  901 

97  953 

02  0^7 

85  948 

21 

40 

83  914 

97  978 

02  022 

85  936 

20 

41 

83  927 

98  003 

01997 

85  924 

19 

42 

83  940 

98  029 

01971 

85  912 

18 

43 

83  954 

98  054 

01946 

85  900 

17 

44 

83  967 

98  079 

01921 

85  888 

16 

45 

83  980 

98104 

01896 

85  876 

15 

46 

83  993 

98130 

01870 

85  864 

14 

47 

84  006 

98  155 

01845 

85  851 

13 

48 

84  020 

98  180 

01820 

85  839 

12 

49 

84  033 

98  206 

01794 

85  827 

11 

50 

84  046 

98  231 

01769 

85  815 

lO 

51 

84  059 

98  256 

01  744 

85  803 

9 

52 

84  072 

98  281 

01719 

85  791 

8 

53 

84  085 

98  307 

01693 

85  779 

7 

54 

84  098 

98  332 

01668 

85  766 

6 

55 

84112 

98  357 

01643 

85  754 

5 

56 

84125 

98  383 

01617 

85  742 

4 

57 

84138 

98  408 

01592 

85  730 

3 

58 

84  151 

98  433 

01567 

85  718 

2 

59 

84  164 

98  458 

01542 

85  706 

1 

60 

84177 

98  484 

01516 

85  693 

0 

9 

log  cos 

9 

log  cot 

lO 

log  tan 

9 

log  sin 

f 

f 

44° 

49 

f 

log  sin 

log  tan 

log  cot 

log  cos 

f 

9 

84  177 

9 

98  484 

lO 

01516 

9 

85  693 

o 

60 

1 

84190 

98  509 

01491 

85  681 

59 

2 

84  203 

98  534 

01466 

85  669 

58 

3 

84  216 

98  560 

01440 

85  657 

57 

4 

84  229 

98  585 

01415 

85  645 

56 

5 

84  242 

98  610 

01390 

85  632 

55 

6 

84  255 

98  635 

01365 

85  620 

54 

7 

84  269 

98  661 

01339 

85  608 

53 

8 

84  282 

98  686 

01314 

85  596 

52 

9 

84  295 

98  711 

01289 

85  583 

51 

10 

84  308 

98  737 

01263 

85  571 

50 

11 

84  321 

98  762 

01238 

85  559 

49 

12 

84  334 

98  787 

01213 

85  547 

48 

13 

84  347 

98  812 

01188 

85  534 

47 

14 

84  360 

98  838 

01162 

85  522 

46 

15 

84  373 

98  863 

01137 

85  510 

45 

16 

84  385 

98  888 

01112 

85  497 

44 

17 

84  398 

98  913 

01087 

85  485 

43 

18 

84  411 

98  939 

01061 

85  473 

42 

19 

84  424 

98  964 

01036 

85  460 

41 

20 

84  437 

98  989 

01011 

85  448 

40 

21 

84  450 

99  015 

00  985 

85  436 

39 

22 

84  463 

99  040 

00  960 

85  423 

38 

23 

84  476 

99  065 

00  935 

85  411 

37 

24 

84  489 

99  090 

00  910 

85  399 

36 

25 

84  502 

99116 

00  884 

85  386 

35 

26 

84  515 

99141 

00  859 

85  374 

34 

27 

84  528 

99166 

00  834 

85  361 

33 

28 

84  540 

99191 

00  809 

85  349 

32 

29 

84  553 

99  217 

00  783 

85  337 

31 

30 

84  566 

99  242 

00  758 

85  324 

30 

31 

84  579 

99  267 

00  733 

85  312 

29 

32 

84  592 

99  293 

00  707 

85  299 

28 

ZZ 

84  605 

99  318 

00  682 

85  287 

27 

34 

84  618 

99  343 

00  657 

85  274 

26 

35 

84  630 

99  368 

00  632 

85  262 

25 

36 

84  643 

99  394 

00  606 

85  250 

24 

37 

84  656 

99  419 

00  581 

85  237 

23 

38 

84  669 

99  444 

00  556 

85  225 

22 

39 

84  682 

99  469 

00  531 

85  212 

21 

40 

84  694 

99  495 

00  505 

85  200 

20 

41 

84  707 

99  520 

00  480 

85  187 

19 

42 

84  720 

99  545 

00  455 

85  175 

18 

43 

84  733 

99  570 

00  430 

85  162 

17 

44 

84  745 

99  596 

00  404 

85  150 

16 

45 

84  758 

99  621 

00  379 

85  137 

15 

46 

84  771 

99  646 

00  354 

85  125 

14 

47 

84  784 

99  672 

00  328 

85  112 

13 

48 

84  796 

99  697 

00  303 

85  100 

12 

49 

84  809 

99  722 

00  278 

85  087 

11 

50 

84  822 

99  747 

00  253 

85  074 

10 

51 

84  835 

99  773 

00  227 

85  062 

9 

52 

84  847 

99  798 

00  202 

85  049 

8 

53 

84  860 

99  823 

00177 

85  037 

7 

54 

84  873 

99  848 

00152 

85  024 

6 

55 

84  885 

99  874 

00126 

85  012 

5 

56 

84  898 

99  899 

00101 

84  999 

4 

57 

84  911 

99  924 

00  076 

84  986 

3 

58 

84  923 

99  949 

00  051 

84  974 

2 

59 

84  936 

99  975 

00  025 

84  961 

1 

60 

84  949 

00  000 

00  000 

84  949 

O 

9 

lO 

lO 

9 

f 

log  cos 

log  cot 

log  tan 

log  sin 

r 

46' 


46' 


50 


TABLE   IV. 

^ 

EoR  Determining  with 

Greater  Accuracy  than  can  be  done  by   1 

MEANS  OF  Table  III. : 

1.   log  sin,  log  tan,  and  log  cot,  when  the  angle  is  between  0°  and  2° ; 

2.    log  cos,  log  tan,  and  log  cot,  when  the  angle  is  between  88°  and  90° ; 

3.   The  value  of  the  angle  when  the  logarithm  of  the  function  does  not 

lie  between  the  limits  8.  54  684  and  11.  45  316. 

FORMULAS   FOR  THE  USE   OF  THE   NUMBERS  S   AND   T. 

I.    When  the  angle  a  is  between  0°  and  2° : 

log  sin  a  =  log  a"  -f  S. 

log  a"  =  log  sin  a  -  S, 

log  tan  a  =  log  o"  +  T. 

=  log  tan  a-  T, 

log  cot  a  =  colog  tan  a. 

=  colog  cot  a—  T. 

II.   When  the  angle  a  is  between  88°  and  90° :                         | 

log  cos  a  =  log  (90°  -a)"  +  S. 

log  (90°  -ay  =  log  cos  a-  S, 

log  cot  a  =  log  (90°  -  a)"  +  T. 

=  log  cot  a-  T, 

log  tan  a  =  colog  cot  a. 

=  colog  tan  a—  T, 

and  a  =  90°-  (90°  -  a). 

Values  of  S  aistd  T. 

a"              S            log  sin  a 

a"             T            log  tan  a 

a              T             log  tan  a 

0                                — 

0                                — 

5  146                     8. 39  713 

4.  68  557 

4.68  557 

4.  68  567 

2409                     8.06740 

200                     6.98  660 

5  424                     8.41999 

4. 68  556 

4.  68  558 

4.  68  568 

3  417                     8.  21  920 

1  726                     7.  92  263 

5  689                     8.  44  072 

4.  68  555 

4.68  559 

4.  68  569 

3  823                     8.26  795 

2  432                     8.  07  156 

5  941                      8.  45  955 

4.  68  55i 

4. 68  560 

4.  68  570 

4  190                    8. 30  776 

2  976                     8. 15  924 

6  184                     8.  47  697 

4.  68  554 

4.  68  561 

4.  68  571 

4  840                     8.  37  038 

3  434                      8.  22  142 

6  417                     8.49  305 

4.68  553 

4.  68  562 

4.  68  572 

5  414                      8.41904 

3  838                     8.26  973 

6  642                      8.50  802 

4.  68  552 

4.  68  563 

4.  68  573 

5  932                      8.  45  872 

4  204                     8.30  930 

6  859                      8.  52  200 

4.68  551 

4.  68  564 

4.  68  574 

6  408                      8.  49  223 

4  540                     8.  34  270 

7  070                     8.53  516 

4.68  550 

4.  68  565 

4.68  575 

6  633                     8.50  721 

4  699                     8.  35  766 

7  173                     8.  54  145 

4.  68  550 

4.  68  565 

4.  68  575 

6  851                     8.52  125 

4  853                     8.37167 

,  7  274                     8.  54  753 

4.  68  549 

4.  68  566 

7  267                     8.  54  684 

5  146                     8.  39  713 

a"              S            log  sin  a 

a"             T            log  tan  a 

a             T            log  tan  a 

TABLE  v.- 

Circumferences  and  Areas  of  Circles,  ^i 

If  ^  =  the  radius  of  the  circle 

,  the  circumference  =  2ir 

N. 

If  iV  =  the  radius  of  the  circle,  the  area 

=  wN\ 

If  iV  =  the  circumference  of  the  circle,  the  radius  =  — 

2  IT 

N. 

If  iV  —  the  circumference  of  the  circle,  the  area     — 

4ir 

N'', 

N 

2'ir^Y 

ir^2 

27r 

47r 

N 

2  7r^ 

TTNi 

2-ir 

4Tr 

O 

0.00 

0.0 

0.000 

0.00 

50 

314. 16 

7  854 

7.96 

198. 94 

1 

6.28 

3.1 

0.159 

0.08 

51 

320.  44 

8171 

8.12 

206. 98 

2 

12.57 

12.6 

0.318 

0.32 

52 

326.  73 

8  495 

8.28 

215. 18 

3 

18.85 

28.3 

0.477 

0.72 

53 

333.  01 

8  825 

8.44 

223. 53 

4 

25. 13 

50.3 

0.637 

1.27 

54 

339.  29 

9161 

8.59 

232.05 

6 

31.42 

78.5 

0.796 

1.99 

55 

345.  58 

9  503 

8.75 

240.  72 

6 

37.70 

113.1 

0.955 

2.86 

56 

351.86 

9  852 

8.91 

249.  55 

7 

43.98 

153.9 

1.114 

3.90 

57 

358. 14 

10  207 

9.07 

258.  55 

8 

50.27 

201.1 

1.273 

5.09 

58 

364.  42 

10  568 

9.23 

267.  70 

9 

56.55 

254.5 

1.432 

6.45 

59 

370.  71 

10  936 

9.39 

277. 01 

10 

62.83 

314.2 

1.592 

7.96 

60 

376.  99 

11310 

9.55 

286. 48 

11 

69.12 

380.1 

1.  751 

9.63 

61 

383.  27 

11690 

9.71 

296. 11 

12 

75.40 

452.4 

1.910 

11.46 

62 

389.  56 

12  076 

9.87 

305.  90 

13 

81.68 

530.9 

2.069 

13.45 

63 

395.  84 

12  469 

10.03 

315.  84 

14 

87.96 

615.8 

2.228 

15.60 

64 

402. 12 

12  868 

10.19 

325.95 

15 

94.25 

706.9 

2.387 

17.90 

65 

408.  41 

13  273 

10.35 

336.  21 

16 

100.  53 

804.2 

2.546 

20.37 

66 

414.  69 

13  685 

10.50 

346.64 

17 

106.  81 

907.9 

2.706 

23.00 

67 

420.  97 

14103 

10.66 

357.22 

18 

113. 10 

1017.9 

2.865 

25.78 

68 

427.  26 

14  527 

10.82 

367.  97 

19 

119.38 

1 134. 1 

3.024 

28.73 

69 

433. 54 

14  957 

10.98 

378.  87 

20 

125.  66 

1  256.  6 

3.183 

31.83 

70 

439.  82 

15  394 

11.14 

389.  93 

21 

131.95 

1  385.  4 

3.342 

35.  09 

71 

446. 11 

15  837 

11.30 

401. 15 

22 

138.  23 

1  520.  5 

3.501 

38.52 

72 

452.  39 

16  286 

11.46 

412.  53 

23 

144.  51 

1  661. 9 

3.661 

42.10 

73 

458.  67 

16  742 

11.62 

424.  07 

24 

150.  80 

1  809.  6 

3.820 

45.84 

74 

464.96 

17  203 

11.78 

435.  77 

25 

157.08 

1  963.  5 

3.979 

49.74 

75 

471.  24 

17  671 

11.94 

447.62 

26 

163.36 

2  123.  7 

4.138 

53.79 

76 

477.  52 

18146 

12.10 

459.64 

27 

169.  65 

2  290.  2 

4.297 

58.01 

77 

483.81 

18  627 

12.25 

471.81 

28 

175.93 

2  463.  0 

4.456 

62.39 

78 

490.09 

19113 

12.41 

484. 15 

29 

182.  21 

2  642. 1 

4.615 

66.92 

79 

496.  37 

19  607 

12.57 

496.  64 

30 

188.  50 

2  827.  4 

4.775 

71.62 

80 

502.65 

20106 

12.73 

509.  30 

31 

194.  78 

3  019. 1 

4.934 

76.47 

81 

508.  94 

20  612 

12.89 

522. 11 

32 

201.  06 

3  217.0 

5.093 

81.49 

82 

515.22 

21124 

13.05 

535.08 

33 

207.35 

3  421.  2 

5.252 

86.66 

83 

521.  50 

21642 

13.21 

548.  21 

34 

213.  63 

3  631.  7 

5.411 

91.99 

84 

527.  79 

22167 

13.37 

561.  50 

35 

219.  91 

3  848.  5 

5.  570 

97.48 

85 

534.  07 

22  698 

13.53 

574.  95 

36 

226. 19 

4  071.5 

5.730 

103. 13 

86 

540.  35 

23  235 

13.69 

588.  55 

37 

232.  48 

4  300.8 

5.889 

108. 94 

87 

546. 64 

23  779 

13.85 

602.  32 

38 

238.  76 

4  536.5 

6.048 

114.91 

88 

552.  92 

24  328 

14.01 

616.  25 

39 

245. 04 

4  778. 4 

6.207 

121.  04 

89 

559.  20 

24  885 

14.16 

630.33 

40 

251.  33 

5  026.  5 

6.366 

127.  32 

90 

565.  49 

25  447 

14.32 

644.  58 

41 

257.  61 

5  281.0 

6.525 

133. 77 

91 

571.77 

26  016 

14.48 

658.  98 

42 

263.89 

5  541.8 

6.685 

140.  37 

92 

578.  05 

26  590 

14.64 

673. 54 

43 

270. 18 

5  80S.  8 

6.844 

147.  14 

93 

584.  34 

27  172 

14.80 

688.  27 

44 

276.  46 

6  082. 1 

7.003 

154. 06 

94 

590.62 

27  759 

14.96 

703. 15 

45 

282.  74 

6  361.7 

7.162 

161. 14 

95 

596.  90 

28  353 

15.12 

718. 19 

46 

289.  03 

6  647.  6 

7.321 

168.  39 

96 

603. 19 

28  953 

15.28 

733.  39 

47 

295.  31 

6  939.  8 

7.480 

175.  79 

97 

609.  47 

29  559 

15.44 

748.  74 

48 

301.  59 

7  238.  2 

7.639 

183. 35 

98 

615.  75 

30172 

15.60 

764.  26 

49 

307.  88 

7  543.  0 

7.799 

191.  07 

99 

622.04 

30  791 

15.76 

779.  94 

50 

314. 16 

2  7riVr 

7  854.0 

7.958 

27r 

198.  94 

47r 

100 

628.  32 

2iriV^ 

31416 

15.92 

27r 

795.  77 

47r 

U 

N 

52 

TABLE  VL  -  MTUEAL  FUNCTIONS 

• 

f 

o° 

1° 

2° 

3° 

4° 

t 

sin 

cos 

sin  cos 

sin 

cos 

sin 

cos 

sin 

COS 

O 

0000 

1.000 

0175  9998 

0349 

9994 

0523 

9986 

0698 

9976 

60 

1 

0003 

1.000 

0177  9998 

0352 

9994 

0526 

9986 

0700 

9975 

59 

2 

0006 

1.000 

0180  9998 

0355 

9994 

0529 

9986 

0703 

9975 

58 

3 

0009 

1.000 

0183  9998 

0358 

9994 

0532 

9986 

0706 

9975 

57 

4 

0012 

1.000 

0186  9998 

0361 

9993 

0535 

9986 

0709 

9975 

56 

5 

0015 

1.000 

0189  9998 

0364 

9993 

0538 

9986 

0712 

9975 

55 

6 

0017 

1.000 

0192  9998 

0366 

9993 

0541 

9985 

0715 

9974 

54 

7 

0020 

1.000 

0195  9998 

0369 

9993 

0544 

9985 

0718 

9974 

53 

8 

0023 

1.000 

0198  9998 

0372 

9993 

0547 

9985 

0721 

9974 

52 

9 

0026 

]..000 

0201  9998 

0375 

9993 

0550 

9985 

0724 

9974 

51 

10 

0029 

1.000 

0204  9998 

0378 

9993 

0552 

9985 

0727 

9974 

50 

11 

0032 

1.000 

0207  9998 

0381 

9993 

0555 

9985 

0729 

9973 

49 

12 

0035 

1.000 

0209  9998 

0384 

9993 

0558 

9984 

0732 

9973 

48 

13 

0038 

1.000 

0212  9998 

0387 

9993 

0561 

9984 

0735 

9973 

47 

14 

0041 

1.000 

0215  9998 

0390 

9992 

0564 

9984 

0738 

9973 

46 

15 

0044 

1.000 

0218  9998 

0393 

9992 

0567 

9984 

0741 

9973 

45 

16 

0047 

1.000 

0221  9998 

0396 

9992 

0570 

9984 

0744 

9972 

44 

17 

0049 

1.000 

0224  9997 

0398 

9992 

0573 

9984 

0747 

9972 

43 

18 

0052 

1.000 

0227  9997 

0401 

9992 

0576 

9983 

0750 

9972 

42 

19 

0055 

1.000 

0230  9997 

0404 

9992 

0579 

9983 

0753 

9972 

41 

20 

0058 

1.000 

0233  9997 

(H07 

9992 

0581 

9983 

0756 

9971 

40 

21 

0061 

1.000 

0236  9997 

0410 

9992 

0584 

9983 

0758 

9971 

39 

22 

0064 

1.000 

0239  9997 

0413 

9991 

0587 

9983 

0761 

9971 

38 

23 

0067 

1.000 

0241  9997 

0416 

9991 

0590 

9983 

0764 

9971 

37 

24 

0070 

1.000 

0244  9997 

0419 

9991 

0593 

9982 

0767 

9971 

36 

25 

0073 

1.000 

0247  9997 

0422 

9991 

0596 

9982 

0770 

9970 

35 

26 

0076 

1.000 

0250  9997 

0425 

9991 

0599 

9982 

0773 

9970 

34 

27 

0079 

1.000 

0253  9997 

0427 

9991 

0602 

9982 

0776 

9970 

33 

28 

0081 

1.000 

0256  9997 

0430 

9991 

0605 

9982 

0779 

9970 

32 

29 

0084 

1.000 

0259  9997 

0433 

9991 

0608 

9982 

0782 

9969 

31 

30 

0087 

1.000 

0262  9997 

0436 

9990 

0610 

9981 

0785 

9969 

30 

31 

0090 

1.000 

0265  9996 

0439 

9990 

0613 

9981 

0787 

9969 

29 

32 

0093 

1.000 

0268  9996 

0442 

9990 

0616 

9981 

0790 

9969 

28 

33 

0096 

1.000 

0270  9996 

0445 

9990 

0619 

9981 

0793 

9968 

27 

34 

0099 

1.000 

0273  9996 

0448 

9990 

0622 

9981 

0796 

9968 

26 

35 

0102 

9999 

0276  9996 

0451 

9990 

0625 

9980 

0799 

9968 

25 

36 

0105 

9999 

0279  9996 

0454 

9990 

0628 

9980 

0802 

9968 

24 

37 

0108 

9999 

0282  9996 

0457 

9990 

0631 

9980 

0805 

9968 

23 

38 

0111 

9999 

0285  9996 

0459 

9989 

0634 

9980 

0808 

9967 

22 

39 

0113 

9999 

0288  9996 

0462 

9989 

0637 

9980 

0811 

9967 

21 

40 

0116 

9999 

0291  9996 

0465 

9989 

0640 

9980 

0814 

9967 

20 

41 

0119 

9999 

0294  9996 

0468 

9989 

0642 

9979 

0816 

9967 

19 

42 

0122 

9999 

0297  9996 

0471 

9989 

0645 

9979 

0819 

9966 

18 

43 

0125 

9999 

0300  9996 

(H74 

9989 

0648 

9979 

0822 

9966 

17 

44 

0128 

9999 

0302  9995 

0477 

9989 

0651 

9979 

0825 

9966 

16 

45 

0131 

9999 

0305  9995 

0480 

9988 

0654 

9979 

0828 

9966 

15 

46 

0134 

9999 

0308  9995 

0483 

9988 

0657 

9978 

0831 

9965 

14 

47 

0137 

9999 

0311  9995 

0486 

9988 

0660 

9978 

0834 

9965 

13 

48 

0140 

9999 

0314  9995 

0488 

9988 

0663 

9978 

0837 

9965 

12 

49 

0143 

9999 

0317  9995 

0491 

9988 

0666 

9978 

0840 

9965 

11 

50 

0145 

9999 

0320  9995 

0494 

9988 

0669 

9978 

0843 

9964 

10 

51 

0148 

9999 

0323  9995 

0497 

9988 

0671 

9977 

0845 

9964 

9 

52 

0151 

9999 

0326  9995 

0500 

9987 

0674 

9977 

0848 

9964 

8 

53 

0154 

9999 

0329  9995 

0503 

9987 

0677 

9977 

0851 

9964 

7 

54 

0157 

9999 

0332  9995 

0506 

9987 

0680 

9977 

0854 

9963 

6 

55 

0160 

9999 

0334  9994 

0509 

9987 

0683 

9977 

0857 

9963 

5 

56 

0163 

9999 

0337  9994 

0512 

9987 

0686 

9976 

0860 

9963 

4 

57 

0166 

9999 

0340  9994 

0515 

9987 

0689 

9976 

0863 

9963 

3 

58 

0169 

9999 

0343  9994 

0518 

9987 

0692 

9976 

0866 

9962 

2 

59 

0172 

9999 

0346  9994 

0520 

9986 

0695 

9976 

0869 

9962 

1 

60 

0175 

9999 

0349  9994 

0523 

9986 

0698 

9976 

0872 

9962 

O 

cos   sin 
89° 

cos  sin 
88° 

cos 

sin 

cos  sin 
86° 

cos 

8; 

sin 

>° 

r 

87° 

f 

'  NATURAL 

SINES  AND 

COSINES. 

53 

/ 

5° 

6° 

7° 

8° 

9° 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

0872 

9962 

1045 

9945 

1219 

9925 

1392 

9903 

1564 

9877 

60 

1 

0874 

9962 

1048 

9945 

1222 

9925 

1395 

9902 

1567 

9876 

59 

2 

0877 

9961 

1051 

9945 

1224 

9925 

1397 

9902 

1570 

9876 

58 

3 

0880 

9961 

1054 

9944 

1227 

9924 

1400 

9901 

1573 

9876 

57 

4 

0883 

9961 

1057 

9944 

1230 

9924 

1403 

9901 

1576 

9875 

56 

5 

0886 

9961 

1060 

9944 

1233 

9924 

1406 

9901 

1579 

9875 

55 

6 

0889 

9960 

1063 

9943 

1236 

9923 

1409 

9900 

1582 

9874 

54 

7 

0892 

9960 

1066 

9943 

1239 

9923 

1412 

9900 

1584 

9874 

53 

8 

0895 

9960 

1068 

9943 

1241 

9923 

1415 

9899 

1587 

9873 

52 

9 

0898 

9960 

1071 

9942 

1245 

9922 

1418 

9899 

1590 

9873 

51 

10 

0901 

9959 

1074 

9942 

1248 

9922 

1421 

9899 

1593 

9872 

50 

11 

0903 

9959 

1077 

9942 

1250 

9922 

1423 

9898 

1596 

9872 

49 

12 

0906 

9959 

1080 

9942 

1253 

9921 

1426 

9898 

1599 

9871 

48 

13 

0909 

9959 

1083 

9941 

1256 

9921 

1429 

9897 

1602 

9871 

47 

14 

0912 

9958 

1086 

9941 

1259 

9920 

1432 

9897 

1605 

9870 

46 

15 

0915 

9958 

1089 

9941 

1262 

9920 

1435 

9897 

1607 

9870 

45 

16 

0918 

9958 

1092 

9940 

1265 

9920 

1438 

9896 

1610 

9869 

44 

17 

0921 

9958 

1094 

9940 

1268 

9919 

1441 

9896 

1613 

9869 

43 

18 

0924 

9957 

1097 

9940 

1271 

9919 

1444 

9895 

1616 

9869 

42 

19 

0927 

9957 

1100 

9939 

1274 

9919 

1446 

9895 

1619 

9868 

41 

20 

0929 

9957 

1103 

9939 

1276 

9918 

1449 

9894 

1622 

9868 

40 

21 

0932 

9956 

1106 

9939 

1279 

9918 

1452 

9894 

1625 

9867 

39 

22 

0935 

9956 

1109 

9938 

1282 

9917 

1455 

9894 

1628 

9867 

38 

23 

0938 

9956 

1112 

9938 

1285 

9917 

1458 

9893 

1630 

9866 

37 

24 

0941 

9956 

1115 

9938 

1288 

9917 

1461 

9893 

1633 

9866 

36 

25 

0944 

9955 

1118 

9937 

1291 

9916 

1464 

9892 

1636 

9865 

35 

26 

0947 

9955 

1120 

9937 

1294 

9916 

1467 

9892 

1639 

9865 

34 

27 

0950 

9955 

1123 

9937 

1297 

9916 

1469 

9891 

1642 

9864 

33 

28 

0953 

9955 

1126 

9936 

1299 

9915 

1472 

9891 

1645 

9864 

32 

29 

0956 

9954 

1129 

9936 

1302 

9915 

1475 

9891 

1648 

9863 

31 

30 

0958 

9954 

1132 

9936 

1305 

9914 

1478 

9890 

1650 

9863 

30 

31 

0961 

9954 

1135 

9935 

1308 

9914 

1481 

9890 

1653 

9862 

29 

32 

0964 

9953 

1138 

9935 

1311 

9914 

1484 

9889 

1656 

9862 

28 

33 

0967 

9953 

1141 

9935 

1314 

9913 

1487 

9889 

1659 

9861 

27 

34 

0970 

9953 

1144 

9934 

1317 

9913 

1490 

9888 

1662 

9861 

26 

35 

0973 

9953 

1146 

9934 

1320 

9913 

1492 

9888 

1665 

9860 

25 

36 

0976 

9952 

1149 

9934 

1323 

9912 

1495 

9888 

1668 

9860 

24 

37 

0979 

9952 

1152 

9933 

1325 

9912 

1498 

9887 

1671 

9859 

23 

38 

0982 

9952 

1155 

9933 

1328 

9911 

1501 

9887 

1673 

9859 

22 

39 

0985 

9951 

1158 

9933 

1331 

9911 

1504 

9886 

1676 

9859 

21 

40 

0987 

9951 

1161 

9932 

1334 

9911 

1507 

9886 

1679 

9858 

20 

41 

0990 

9951 

1164 

9932 

1337 

9910 

1510 

9885 

1682 

9858 

19 

42 

0993 

9951 

1167 

9932 

1340 

9910 

1513 

9885 

1685 

9857 

18 

43 

0996 

9950 

1170 

9931 

1343 

9909 

1515 

9884 

1688 

9857 

17 

44 

0999 

9950 

1172 

9931 

1346 

9909 

1518 

9884 

1691 

9856 

16 

45 

1002 

9950 

1175 

9931 

1349 

9909 

1521 

9884 

1693 

9856 

15 

46 

1005 

9949 

1178 

9930 

1351 

9908 

1524 

9883 

1696 

9855 

14 

47 

1008 

9949 

1181 

9930 

1354 

9908 

1527 

9883 

1699 

9855 

13 

48 

1011 

9949 

1184 

9930 

1357 

9907 

1530 

9882 

1702 

9854 

12 

49 

1013 

9949 

1187 

9929 

1360 

9907 

1533 

9882 

1705 

9854 

11 

50 

1016 

9948 

1190 

9929 

1363 

9907 

1536 

9881 

1708 

9853 

lO 

51 

1019 

9948 

1193 

9929 

1366 

9906 

1538 

9881 

1711 

9853 

9 

52 

1022 

9948 

1196 

9928 

1369 

9906 

1541 

9880 

1714 

9852 

8 

53 

1025 

9947 

1198 

9928 

1372 

9905 

1544 

9880 

1716 

9852 

7 

54 

1028 

9947 

1201 

9928 

1374 

9905 

1547 

9880 

1719 

9851 

6 

55 

1031 

9947 

1204 

9927 

1377 

9905 

1550 

9879 

1722 

9851 

5 

56 

1034 

9946 

1207 

9927 

1380 

9904 

1553 

9879 

1725 

9850 

4 

57 

1037 

9946 

1210 

9927 

1383 

9904 

1556 

9878 

1728 

9850 

3 

58 

1039 

9946 

1213 

9926 

1386 

9903 

1559 

9878 

1731 

9849 

2 

59 

1042 

9946 

1216 

9926 

1389 

9903 

1561 

9877 

1734 

9849 

1 

60 

1045 

9945 

1219 

9925 

1392 

9903 

1564 

9877 

1736 

9848 

0 

cos  sin 
84° 

cos  sin 
83° 

cos  sin 
82° 

cos  sin 
81° 

cos  sin 
80° 

f 

f 

54  NATURAL   SINES   AND   COSINES. 


r 

io° 

11° 

12° 

13° 

14° 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

O 

1736 

9848 

1908 

9816 

2079 

9781 

2250 

9744 

2419 

9703 

60 

1 

1739 

9848 

1911 

9816 

2082 

9781 

2252 

9743 

2422 

9702 

59 

2 

1742 

9847 

1914 

9815 

2085 

9780 

2255 

9742 

2425 

9702 

58 

3 

1745 

9847 

1917 

9815 

2088 

9780 

2258 

9742 

2428 

9701 

57 

4 

1748 

9846 

1920 

9814 

2090 

9779 

2261 

9741 

2431 

9700 

56 

6 

1751 

9846 

1922 

9813 

2093 

9778 

2264 

9740 

2433 

9699 

55 

6 

1754 

9845 

1925 

9813 

2096 

9778 

2267 

9740 

2436 

9699 

54 

7 

1757 

9845 

1928 

9812 

2099 

9777 

2269 

9739 

2439 

9698 

53 

8 

1759 

9844 

1931 

9812 

2102 

9777 

2272 

9738 

2442 

9697 

52 

9 

1762 

9843 

1934 

9811 

2105 

9776 

2275 

9738 

2445 

9697 

51 

lO 

1765 

9843 

1937 

9811 

2108 

9775 

2278 

9737 

2447 

9696 

50 

11 

1768 

9842 

1939 

9810 

2110 

9775 

2281 

9736 

2450 

9695 

49 

12 

1771 

9842 

1942 

9810 

2113 

9774 

2284 

9736 

2453 

9694 

48 

13 

1774 

9841 

1945 

9809 

2\\^  9774 

2286 

9735 

2456 

9694 

47 

14 

1777 

9841 

1948 

9808 

2119 

9773 

2289 

9734 

2459 

9693 

46 

15 

1779 

9840 

1951 

9808 

2122 

9772 

2292 

9734 

2462 

9692 

45 

16 

1782 

9840 

1954 

9807 

2125 

9772 

2295 

9733 

2464 

9692 

44 

17 

1785 

9839 

1957 

9807 

2127 

9771 

2298 

9732 

2467 

9691 

43 

18 

1788 

9839 

1959 

9806 

2130 

9770 

2300 

9732 

2470 

9690 

42 

19 

1791 

9838 

1962 

9806 

2133 

9770 

2303 

9731 

2473 

9689 

41 

20 

1794 

9838 

1965 

9805 

2136 

9769 

2306 

9730 

2476 

9689 

40 

21 

1797 

9837 

1968 

980+ 

2139 

9769 

2309 

9730 

2478 

9688 

39 

22 

1799 

9837 

1971 

9804 

2142 

9768 

2312 

9729 

2481 

9687 

38 

23 

1802 

9836 

1974 

9803 

2145 

9767 

2315 

9728 

2484 

9687 

37 

24 

1805 

9836 

1977 

9803 

2147 

9767 

2317 

9728 

2487 

9686 

36 

25 

1808 

9835 

1979 

9802 

2150 

9766 

2320 

9727 

2490 

9685 

35 

26 

1811 

9835 

1982 

9802 

2153 

9765 

2323 

9726 

2493 

9684 

34 

27 

1814 

9834 

1985 

9801 

2156 

9765 

2326 

9726 

2495 

9684 

33 

28 

1817 

9834 

1988 

9800 

2159 

9764 

2329 

9725 

2498 

9683 

32 

29 

1819 

9833 

1991 

9800 

2162 

9764 

2332 

9724 

2501 

9682 

31 

30 

1822 

9833 

1994 

9799 

2164 

9763 

2334 

9724 

2504 

9681 

30 

31 

1825 

9832 

1997 

9799 

2167 

9762 

2337 

9723 

2507 

9681 

29 

32 

1828 

9831 

1999 

9798 

2170 

9762 

2340 

9722 

2509 

9680 

28 

33 

1831 

9831 

2002 

9798 

2173 

9761 

2343 

9722 

2512 

9679 

27 

34 

1834 

9830 

2005 

9797 

2176 

9760 

2346 

9721 

2515 

9679 

26 

35 

1837 

9830 

2008 

9796 

2179 

9760 

2349 

9720 

2518 

9678 

25 

36 

1840 

9829 

2011 

9796 

2181 

9759 

2351 

9720 

2521 

9677 

24 

37 

1842 

9829 

2014 

9795 

2184 

9759 

2354 

9719 

2524 

9676 

23 

38 

1845 

9828 

2016 

9795 

2187 

9758 

2357 

9718 

2526 

9676 

22 

39 

1848 

9828 

2019 

9794 

2190 

9757 

2360 

9718 

2529 

9675 

21 

40 

1851 

9827 

2022 

9793 

2193 

9757 

2363 

9717 

2532 

9674 

20 

41 

1854 

9827 

2025 

9793 

2196 

9756 

2366 

9716 

2535 

9673 

19 

42 

1857 

9826 

2028 

9792 

2198 

9755 

2368 

9715 

2538 

9673 

18 

43 

1860 

9826 

2031 

9792 

2201 

9755 

2371 

9715 

2540 

9672 

17 

44 

1862 

9825 

2034 

9791 

2204 

9754 

2374 

9714 

2543 

9671 

16 

45 

1865 

9825 

2036 

9790 

2207 

9753 

2377 

9713 

2546 

9670 

15 

46 

1868 

9824 

2039 

9790 

2210 

9753 

2380 

9713 

2549 

9670 

14 

47 

1871 

9823 

2042 

9789 

2213 

9752 

2383 

9712 

2552 

9669 

13 

48 

1874 

9823 

2045 

9789 

2215 

9751 

2385 

9711 

2554 

9668 

12 

49 

1877 

9822 

2048 

9788 

2218 

9751 

2388 

9711 

2557 

9667 

11 

50 

1880 

9822 

2051 

9787 

2221 

9750 

2391 

9710 

2560 

9667 

10 

51 

1882 

9821 

2054 

9787 

2224 

9750 

2394 

9709 

2563 

9666 

9 

52 

1885 

9821 

2056 

9786 

2227 

9749 

2397 

9709 

2566 

9665 

8 

S3 

1888 

9820 

2059 

9786 

2230 

9748 

2399 

9708 

2569 

9665 

7 

54 

]891 

9820 

2062 

9785 

2233 

9748 

2402 

9707 

2571 

9664 

6 

55 

1894 

9819 

2065 

9784 

2235 

9747 

2405 

9706 

2574 

9663 

5 

56 

1897 

9818 

2068 

9784 

2238 

9746 

2408 

9706 

2577 

9662 

4 

57 

1900 

9818 

2071 

9783 

2241 

9746 

2411 

9705 

2580 

9662 

3 

58 

1902 

9817 

2073 

9783 

2244 

9745 

2414 

9704 

2583 

9661 

2 

59 

1905 

9817 

2076 

9782 

2247 

9744 

2416 

9704 

2585 

9660 

1 

60 

1908 

9816 

2079 

9781 

2250 

9744 

2419 

9703 

2588 

9659 

O 

cos  sin 
79° 

cos  sin 
78° 

cos  sin 

77° 

cos 

sin 

cos   sin 

75° 

r 

76° 

f 

NATUKAL   SINES   AND   COSINES. 


55 


f 

15° 

sin  cos 

16° 

17° 

18° 

19° 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

2588 

9659 

2756 

9613 

2924 

9563 

3090 

9511 

3256 

9455 

60 

] 

2591 

9659 

2759 

9612 

2926 

9562 

3093 

9510 

3258 

9454 

59 

2 

2594 

9658 

2762 

9611 

2929 

9561 

3096 

9509 

3261 

9453 

58 

3 

2597 

9657 

2165 

9610 

2932 

9560 

3098 

9508 

3264 

9452 

57 

4 

2599 

9656 

2768 

9609 

2935 

9560 

3101 

9507 

3267 

9451  . 

56 

5 

2602 

9655 

2770 

9609 

2938 

9559 

3104 

9506 

3269 

9450 

55 

6 

2605 

9655 

2773 

9608 

2940 

9558 

3107 

9505 

3272 

9449 

54 

7 

2608 

9654 

2776 

9607 

2943 

9557 

3110 

9504 

3275 

9449 

53 

8 

2611 

9653 

2779 

9606 

2946 

9556 

3112 

9503 

3278 

9448 

52 

9 

2613 

9652 

2782 

9605 

2949 

9555 

3115 

9502 

3280 

9447 

51 

lO 

2616 

9652 

2784 

9605 

2952 

9555 

3118 

9502 

3283 

9446 

50 

11 

2619 

9651 

2787 

9604 

2954 

9554 

3121 

9501 

3286 

9445 

49 

12 

2622 

9650 

2790 

9603 

2957 

9553 

3123 

9500 

3289 

9444 

48 

13 

2625 

9649 

2793 

9602 

2960 

9552 

3126 

9499 

3291 

9443 

47 

14 

2628 

9649 

2795 

9601 

2963 

9551 

3129 

9498 

3294 

9442 

46 

15 

2630 

9648 

2798 

9600 

2965 

9550 

3132 

9497 

3297 

9441 

45 

16 

2633 

9647 

2801 

9600 

2968 

9549 

3134 

9496 

3300 

9440 

44 

17 

2636 

9646 

2804 

9599 

2971 

9548 

3137 

9495 

3302 

9439 

43 

18 

2639 

9646 

2807 

9598 

2974 

9548 

3140 

9494 

3305 

9438 

42 

19 

2642 

9645 

2809 

9597 

2977 

9547 

3143 

9493 

3308 

9437 

41 

20 

2644 

9644 

2812 

9596 

2979 

9546 

3145 

9492 

3311 

9436 

40 

21 

2647 

9643 

2815 

9596 

2982 

9545 

3148 

9492 

3313 

9435 

39 

22 

2650 

9642 

2818 

9595 

2985 

9544 

3151 

9491 

3316 

9434 

38 

23 

2653 

9642 

2821 

9594 

2988 

9543 

3154 

9490 

3319 

9433 

37 

24 

2656 

9641 

2823 

9593 

2990 

9542 

3156 

9489 

3322 

9432 

36 

25 

2658 

9640 

2826 

9592 

2993 

9542 

3159 

9488 

3324 

9431 

35 

26 

2661 

9639 

2829 

9591 

2996 

9541 

3162 

9487 

3327 

9430 

34 

27 

2664 

9639 

2832 

9591 

2999 

9540 

3165 

9486 

3330 

9429 

33 

28 

2667 

9638 

2835 

9590 

3002 

9539 

3168 

9485 

3333 

9428 

32 

29 

2670 

9637 

2837 

9589 

3004 

9538 

3170 

9484 

3335 

9427 

31 

30 

2672 

9636 

2840 

9588 

3007 

9537 

3173 

9483 

3338 

9426 

30 

31 

2675 

9636 

2843 

9587 

3010 

9536 

3176 

9482 

3341 

9425 

29 

32 

2678 

9635 

2846 

9587 

3013 

9535 

3179 

9481 

3344 

9424 

28 

ZZ 

2681 

9634 

2849 

9586 

3015 

9535 

3181 

9480 

3346 

9423 

27 

34 

2684 

9633 

2851 

9585 

3018 

9534 

3184 

9480 

3349 

9423 

26 

35 

2686 

9632 

2854 

9584 

3021 

9533 

3187 

9479 

3352 

9422 

25 

36 

2689 

9632 

2857 

9583 

3024 

9532 

3190 

9478 

3355 

9421 

24 

37 

2692 

9631 

2860 

9582 

3026 

9531 

3192 

9477 

3357 

9420 

23 

38 

2695 

9630 

2862 

9582 

3029 

9530 

3195 

9476 

3360 

9419 

22 

39 

2698 

9629 

2865 

9581 

3032 

9529 

3198 

9475 

3363 

9418 

21 

40 

2700 

9628 

2868 

9580 

3035 

9528 

3201 

9474 

3365 

9417 

20 

41 

2703 

9628 

2871 

9579 

3038 

9527 

3203 

9473 

3368 

9416 

19 

42 

2706 

9627 

2874 

9578 

3040 

9527 

3206 

9472 

3371 

9415 

18 

43 

2709 

9626 

2876 

9577 

3043 

9526 

3209 

9471 

3374 

9414 

17 

44 

2712 

9625 

2879 

9577 

3046 

9525 

3212 

9470 

3376 

9413 

16 

45 

2714 

9625 

2882 

9576 

3049 

9524 

3214 

9469 

3379 

9412 

15 

46 

2717 

9624 

2885 

9575 

3051 

9523 

3217 

9468 

3382 

9411 

14 

47 

2720 

9623 

2888 

9574 

3054 

9522 

3220 

9467 

3385 

9410 

13 

48 

2723 

9622 

2890 

9573 

3057 

9521 

3223 

9466 

3387 

9409 

12 

49 

2726 

9621 

2893 

9572 

3060 

9520 

3225 

9466 

3390 

9408 

11 

50 

2728 

9621 

2896 

9572 

3062 

9520 

3228 

9465 

3393 

9407 

lO 

51 

2731 

9620 

2899 

9571 

3065 

9519 

3231 

9464 

3396 

9406 

9 

52 

2734 

9619 

2901 

9570 

3068 

9518 

3234 

9463 

3398 

9405 

8 

53 

2737 

9618 

2904 

9569 

3071 

9517 

3236 

9462 

3401 

9404 

7 

54 

2740 

9617 

2907 

9568 

3074 

9516 

3239 

9461 

3404 

9403 

6 

55 

2742 

9617 

2910 

9567 

3076 

9515 

3242 

9460 

3407 

9402 

5 

56 

2745 

9616 

2913 

9566 

3079 

9514 

3245 

9459 

3409 

9401 

4 

57 

2748 

9615 

2915 

9566 

3082 

9513 

3247 

9458 

3412 

9400 

3 

58 

2751 

9614 

2918 

9565 

3085 

9512 

3250 

9457 

3415 

9399 

2 

59 

2754 

9613 

2921 

9564 

3087 

9511 

3253 

9456 

3417 

9398 

1 

60 

2756 

9613 

2924 

9563- 

3090 

9511 

3256 

9455 

3420 

9397 

O 

cos  sin 

74° 

cos  sin 
73° 

cos  sin 

72° 

cos  sin 
71° 

cos   sin 
70° 

r 

f 

56 


NATURAL   SINES   AND   COSINES. 


f 

20° 

21° 

22° 
sin  cos 

23° 

sin  cos 

24° 
sin  cos 

f 

sin 

cos 

sin 

cos 

o 

3420 

9397 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

60 

1 

3423 

9396 

3586 

9335 

3749 

9271 

3910 

9204 

4070 

9134 

59 

2 

3426 

9395 

3589 

9334 

3751 

9270 

3913 

9203 

4073 

9133 

58 

3 

3428 

9394 

3592 

9333 

3754 

9269 

3915 

9202 

4075 

9132 

57  , 

4 

3431 

9393 

3595 

9332 

3757 

9267 

3918 

9200 

4078 

9131 

56 

5 

3434 

9392 

3597 

9331 

3760 

9266 

3921 

9199 

4081 

9130 

55 

6 

3437 

9391 

3600 

9330 

3762 

9265 

3923 

9198 

4083 

9128 

54 

7 

3439 

9390 

3603 

9328 

3765 

9264 

3926 

9197 

4086 

9127 

53 

8 

3442 

9389 

3605 

9327 

3768 

9263 

3929 

9196 

4089 

9126 

52 

9 

3445 

9388 

3608 

9326 

3770 

9262 

3931 

9195 

4091 

9125 

51 

lO 

3448 

9387 

361  i 

9325 

3773 

9261 

3934 

9194 

4094 

9124 

50 

11 

3450 

9386 

3614 

9324 

3776 

9260 

3937 

9192 

4097 

9122 

49 

12 

3453 

9385 

3616 

9323 

3778 

9259 

3939 

9191 

4099 

9121 

48 

13 

3456 

9384 

3619 

9322 

3781 

9258 

3942 

9190 

4102 

9120 

47 

14 

3458 

9383 

3622 

9321 

3784 

9257 

3945 

9189 

4105 

9119 

46 

15 

3461 

9382 

3624 

9320 

3786 

9255 

3947 

9188 

4107 

9118 

45 

16 

3464 

9381 

3627 

9319 

3789 

9254 

3950 

9187 

4110 

9116 

44 

17 

3467 

9380 

3630 

9318 

3792 

9253 

3953 

9186 

4112 

9115 

43 

18 

3469 

9379 

3633 

9317 

3795 

9252 

3955 

9184 

4115 

9114 

42 

19 

3472 

9378 

3635 

9316 

3797 

9251 

3958 

9183 

4118 

9113 

41 

20 

3475 

9377 

3638 

9315 

3800 

9250 

3961 

9182 

4120 

9112 

40 

21 

3478 

9376 

3641 

9314 

3803 

9249 

3963 

9181 

4123 

9110 

39 

22 

3480 

9375 

3643 

9313 

3805 

9248 

3966 

9180 

4126 

9109 

38 

23 

3483 

9374 

3646 

9312 

3808 

9247. 

3969 

9179 

4128 

9108 

37 

24 

3486 

9373 

3649 

9311 

3811 

9245 

3971 

9178 

4131 

9107 

36 

25 

3488 

9372 

3651 

9309 

3813 

9244 

3974 

9176 

4134 

9106 

35 

26 

3491 

9371 

3654 

9308 

3816 

9243 

3977 

9175 

4136 

9104 

34 

27 

3494 

9370 

3657 

9307 

3819 

9242 

3979 

9174 

4139 

9103 

33 

28 

3497 

9369 

3660 

9306 

3821 

9241 

3982 

9173 

4142 

9102 

32 

29 

3499 

9368 

3662 

9305 

3824 

9240 

3985 

9172 

4144 

9101 

31 

30 

3502 

9367 

3665 

9304 

3827 

9239 

3987 

9171 

4147 

9100 

30 

31 

3505 

9366 

3668 

9303 

3830 

9238 

3990 

9169 

4150 

9098 

29 

32 

3508 

9365 

3670 

9302 

3832 

9237 

3993 

9168 

4152 

9097 

28 

33 

3510 

9364 

3673 

9301 

3835 

9235 

3995 

9167 

4155 

9096 

27 

34 

3513 

9363 

3676 

9300 

3838 

9234 

3998 

9166 

4158 

9095 

26 

35 

3516 

9362 

3679 

9299 

3840 

9233 

4001 

9165 

4160 

9094 

25 

36 

3518 

9361 

3681 

9298 

3843 

9232 

4003 

9164 

4163 

9092 

24 

37 

3521 

9360 

3684 

9297 

3846 

9231 

4006 

9162 

4165 

9091 

23 

38 

3524 

9359 

3687 

9296 

3848 

9230 

4009 

9161 

4168 

9090 

22 

39 

3527 

9358 

3689 

9295 

3851 

9229 

4011 

9160 

4171 

9088 

21 

40 

3529 

9356 

3692 

9293 

3854 

9228 

4014 

9159 

4173 

9088 

20 

41 

3532 

9355 

3695 

9292 

3856 

9227 

4017 

9158 

4176 

9086 

19 

42 

3535 

9354 

3697 

9291 

3859 

9225 

4019 

9157 

4179 

9085 

18 

43 

3537 

9353 

3700 

9290 

3862 

9224 

4022 

9155 

4181 

9084 

17 

44 

3540 

9352 

3703 

9289 

3864 

9223 

4025 

9154 

4184 

9083 

16 

45 

3543 

9351 

3706 

9288 

3867 

9222 

4027 

9153 

4187 

9081 

15 

46 

3546 

9350 

3708 

9287 

3870 

9221 

4030 

9152 

4189 

9080 

14 

47 

3548 

9349 

3711 

9286 

3872 

9220 

4033 

9151 

4192 

9079 

13 

48 

3551 

9348 

3714 

9285 

3875 

9219 

4035 

9150 

4195 

9078 

12 

49 

3554 

9347 

3716 

9284 

3878 

9218 

4038 

9148 

4197 

9077 

11 

50 

3557 

9346 

3719 

9283 

3881 

9216 

4041 

9147 

4200 

9075 

lO 

51 

3559 

9345 

3722 

9282 

3883 

9215 

4043 

9146 

4202 

9074 

9 

52 

3562 

9344 

3724 

9281 

3886 

9214 

4046 

9145 

4205 

9073 

8 

53 

3565 

9343 

3727 

9279 

3889 

9213 

4049 

9144 

4208 

9072 

7 

54 

3567 

9342 

3730 

9278 

3891 

9212 

4051 

9143 

4210 

9070 

6 

55 

3570 

9341 

3733 

9277 

3894 

9211 

4054 

9141 

4213 

9069 

5 

56 

3573 

9340 

3735 

9276 

3897 

9210 

4057 

9140 

4216 

9068 

4 

57 

3576 

9339 

3738 

9275 

3899 

9208 

4059 

9139 

4218 

9067 

3 

58 

3578 

9338 

3741 

9274 

3902 

9207 

4062 

9138 

4221 

9066 

2 

59 

3581 

9337 

3743 

9273 

3905 

9206 

4065 

9137 

4224 

9064 

1 

60 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

4226 

9063 

0 

cos   sin 
69° 

cos  sin 
68° 

cos 

sin 

cos  sin 
66° 

cos 

sin 

f 

67° 

65° 

f 

NATURAL 

SINES  AND 

COSINES. 

57 

/ 

25° 

26° 

sin  cos 

27° 

28° 
sin  cos 

29° 

f 

sin 

cos 

sin 

cos 

sin 

cos 

o 

4226 

9063 

4384 

8988 

4540 

8910 

4695 

8829 

4848 

8746 

60 

1 

4229 

9062 

4386 

8987 

4542 

8909 

4697 

8828 

4851 

8745 

59 

2 

4231 

9061 

4389 

8985 

4545 

8907 

4700 

8827 

4853 

8743 

58 

3 

4234 

9059 

4392 

8984 

4548 

8906 

4702 

8825 

4856 

8742 

57 

4 

4237 

9058 

4394 

8983 

4550 

8905 

4705 

8824 

4858 

8741 

56 

5 

4239 

9057 

4397 

8982 

4553 

8903 

4708 

8823 

4861 

8739 

55 

6 

4242 

9056 

4399 

8980 

4555 

8902 

4710 

8821 

4863 

8738 

54 

7 

4245 

9054 

4402 

8979 

4558 

8901 

4713 

8820 

4866 

8736 

53 

8 

4247 

9053 

4405 

8978 

4561 

8899 

4715 

8819 

4868 

8735 

52 

9 

4250 

9052 

4407 

8976 

4563 

8898 

4718 

8817 

4871 

8733 

51 

lO 

4253 

9051 

4410 

8975 

4566 

8897 

4720 

8816 

4874 

8732 

50 

11 

4255 

9050 

4412 

8974 

4568 

8895 

4723 

8814 

4876 

8731 

49 

12 

4258 

9048 

4415 

8973 

4571 

8894 

4726 

8813 

4879 

8729 

48 

13 

4260 

9047 

4418 

8971 

4574 

8893 

4728 

8812 

4881 

8728 

47 

14 

4263 

9046 

4420 

8970 

4576 

8892 

4731 

8810 

4884 

8726 

46 

15 

4266 

9045 

4423 

8969 

4579 

8890 

4733 

8809 

4886 

8725 

45 

16 

4268 

9043 

4425 

8967 

4581 

8889 

4736 

8808 

4889 

8724 

44 

17 

4271 

9042 

4428 

8966 

4584 

8888 

4738 

8806 

4891 

8722 

43 

18 

4274 

9041 

4431 

8965 

4586 

8886 

4741 

8805 

4894 

8721 

42 

19 

4276 

9040 

4433 

8964 

4589 

8885 

4743 

8803 

4896 

8719 

41 

20 

4279 

9038 

4436 

8962 

4592 

8884 

4746 

8802 

4899 

8718 

40 

21 

4281 

9037 

4439 

8961 

4594 

8882 

4749 

8801 

4901 

8716 

39 

22 

4284 

9036 

4441 

8960 

4597 

8881 

4751 

8799 

4904 

8715 

38 

23 

4287 

9035 

4444 

8958 

4599 

8879 

4754 

8798 

4907 

8714 

37 

24 

4289 

9033 

4446 

8957 

4602 

8878 

4756 

8796 

4909 

8712 

36 

25 

4292 

9032 

4449 

8956 

4605 

8877 

4759 

8795 

4912 

8711 

35 

26 

4295 

9031 

4452 

8955 

4607 

8875 

4761 

8794 

4914 

8709 

34 

27 

4297 

9030 

4454 

8953 

4610 

8874 

4764 

8792 

4917 

8708 

33 

28 

4300 

9028 

4457 

8952 

4612 

8873 

4766 

8791 

4919 

8706 

32 

29 

4302 

9027 

4459 

8951 

4615 

8871 

4769 

8790 

4922 

8705 

31 

30 

4305 

9026 

4462 

8949 

4617 

8870 

4772 

8788 

4924 

8704 

30 

31 

4308 

9025 

4465 

8948 

4620 

8869 

4774 

8787 

4927 

8702 

29 

32 

4310 

9023 

4467 

8947 

4623 

8867 

4777 

8785 

4929 

8701 

28 

33 

4313 

9022 

4470 

8945 

4625 

8866 

4779 

8784 

4932 

8699 

27 

34 

4316 

9021 

4472 

8944 

4628 

8865 

4782 

8783 

4934 

8698 

26 

35 

4318 

9020 

4475 

8943 

4630 

8863 

4784 

8781 

4937 

8696 

25 

36 

4321 

9018 

4478 

8942 

4633 

8862 

4787 

8780 

4939 

8695 

24 

37 

4323 

9017 

4480 

8940 

4636 

8861 

4789 

8778 

4942 

8694 

23 

38 

4326 

9016 

4483 

8939 

4638 

8859 

4792 

8777 

4944 

8692 

22 

39 

4329 

9015 

4485 

8938 

4641 

8858 

4795 

8776 

4947 

8691 

21 

40 

4331 

9013 

4488 

8936 

4643 

8857 

4797 

8774 

4950 

8689 

20 

41 

4334 

9012 

4491 

8935 

4646 

8855 

4800 

8773 

4952 

8688 

19 

42 

4337 

9011 

4493 

8934 

4648 

8854 

4802 

8771 

4955 

8686 

18 

43 

4339 

9010 

4496 

8932 

4651 

8853 

4805 

8770 

4957 

8685 

17 

44 

4342 

9008 

4498 

8931 

4654 

8851 

4807 

8769 

4960 

8683 

16 

45 

4344 

9007 

4501 

8930 

4656 

8850 

4810 

8767 

4962 

8682 

15 

46 

4347 

9006 

4504 

8928 

4659 

8849 

4812 

8766 

4965 

8681 

14 

47 

4350 

9004 

4506 

8927 

4661 

8847 

4815 

8764 

4967 

8679 

13 

48 

4352 

9003 

4509 

8926 

4664 

8846  ' 

4818 

8763 

4970 

8678 

12 

49 

4355 

9002 

4511 

8925 

4666 

8844 

4820 

8762 

4972 

8676 

11 

50 

4358 

9001 

4514 

8923 

4669 

8843 

4823 

8760 

4975 

8675 

10 

51 

4360 

8999 

4517 

8922 

4672 

8842 

4825 

8759 

4977 

8673 

9 

52 

4363 

8998 

4519 

8921 

4674 

8840 

4828 

8757 

498D 

8672 

8 

53 

4365 

8997 

4522 

8919 

4677 

8839 

4830 

8756 

4982 

8670 

7 

54 

4368 

8996 

4524 

8918 

4679 

8838 

4833 

8755 

4985 

8669 

6 

55 

4371 

8994 

4527 

8917 

4682 

8836 

4835 

8753 

4987 

8668 

5 

56 

4373 

8993 

4530 

8915 

4684 

8835 

4838 

8752 

4990 

8666 

4 

57 

4376 

8992 

4532 

8914 

4687 

8834 

4840 

8750 

4992 

8665 

3 

58 

4378 

8990 

4535 

8913 

4690 

8832 

4843 

8749 

4995 

8663 

2 

59 

4381 

8989 

4537 

8911 

4692 

8831 

4846 

8748 

4997 

8662 

1 

60 

4384 

8988 

4540 

8910 

4695 

8829 

4848 

8746 

5000 

8660 

0 

cos   sin 
64° 

cos  sin 
63° 

cos  sin 
62° 

cos 

sin 

cos   sin 
60° 

f 

61° 

' 

58 

NATURAL 

SINES 

AND 

COSINES. 

/ 

30° 

31° 

32° 

sin  cos 

33° 

sin  cos 

34° 

f 

sin 

cos 

sin 

cos 

sin 

cos 

O 

5000 

8660 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

60 

1 

5003 

8659 

5153 

8570 

5302 

8479 

5449 

8385 

5594 

8289 

59 

2 

5005 

8657 

5155 

8569 

5304 

8477 

5451 

8384 

5597 

8287 

58 

3 

5008 

8656 

5158 

8567 

5307 

8476 

5454 

8382 

5599 

8285 

57 

4 

5010 

8654 

5160 

8566 

5309 

8474 

5456 

8380 

5602 

8284 

56 

5 

5013 

8653 

5163 

8564 

5312 

8473 

5459 

8379 

5604 

8282 

55 

6 

5015 

8652 

5165 

8563 

5314 

8471 

5461 

8377 

5606 

8281 

54 

7 

5018 

8650 

5168 

8561 

5316 

8470 

5463 

8376 

5609 

8279 

53 

8 

5020 

8649 

5170 

8560 

5319 

8468 

5466 

8374 

5611 

8277 

52 

9 

5023 

8647 

5173 

8558 

5321 

8467 

5468 

8372 

5614 

8276 

51 

lO 

5025 

8646 

5175 

8557 

5324 

8465 

5471 

8371 

5616 

8274 

50 

11 

5028 

8644 

5178 

8555 

5326 

8463 

5473 

8369 

5618 

8272 

49 

12 

5030 

8643 

5180 

8554 

5329 

8462 

5476 

8368 

5621 

8271 

48 

13 

5033 

8641 

5183 

8552 

5331 

8460 

5478 

8366 

5623 

8269 

47 

14 

5035 

8640 

5185 

8551 

5334 

8459 

5480 

8364 

5626 

8268 

46 

15 

5038 

8638 

5188 

8549 

5336 

8457 

5483 

8363 

5628 

8266 

45 

16 

5040 

8637 

5190 

8548 

5339 

8456 

5485 

8361 

5630 

8264 

44 

17 

5043 

8635 

5193 

8546 

5341 

8454 

5488 

8360 

5633 

8263 

43 

18 

5045 

8634 

5195 

8545 

5344 

8453 

5490 

8358 

5635 

8261 

42 

19 

5048 

8632 

5198 

8543 

5346 

8451 

5493 

8356 

5638 

8259 

41 

20 

5050 

8631 

5200 

8542 

5348 

8450 

5495 

8355 

5640 

8258 

40 

21 

5053 

8630 

5203 

8540 

5351 

8448 

5498 

8353 

5642 

8256 

39 

22 

5055 

8628 

5205 

8539 

5353 

8446 

5500 

8352 

5645 

8254 

38 

23 

5058 

8627 

5208 

8537 

5356 

8445 

5502 

8350 

5647 

8253 

37 

24 

5060 

8625 

5210 

8536 

5358 

8443 

5505 

8348 

5650 

8251 

36 

25 

5063 

8624 

5213 

8534 

5361 

8442 

5507 

8347 

5652 

8249 

35 

26 

5065 

8622 

5215 

8532 

5363 

8440 

5510 

8345 

5654 

8248 

34 

27 

5068 

8621 

5218 

8531 

5366 

8439 

5512 

8344 

5657 

8246 

33 

28 

5070 

8619 

5220 

8529 

5368 

8437 

5515 

8342 

5659 

8245 

32 

29 

5073 

8618 

5223 

8528 

5371 

8435 

5517 

8340 

5662 

8243 

31 

30 

5075 

8616 

5225 

8526 

5373 

8434 

5519 

8339 

5664 

8241 

30 

31 

5078 

8615 

5227 

8525 

5375 

8432 

5522 

8337 

5666 

8240 

29 

32 

5080 

8613 

5230 

8523 

5378 

8431 

5524 

8336 

5669 

8238 

28 

33 

5083 

8612 

5232 

8522 

5380 

8429 

5527 

8334 

5671 

8236 

27 

34 

5085 

8610 

5235 

8520 

5383 

8428 

5529 

8332 

5674 

8235 

26 

35 

5088 

8609 

5237 

8519 

5385 

8426 

5531 

8331 

5676 

8233 

25 

36 

5090 

8607 

5240 

8517 

5388 

8425 

5534 

8329 

5678 

8231 

24 

37 

5093 

8606 

5242 

8516 

5390 

8423 

5536 

8328 

5681 

8230 

23 

38 

5095 

8604 

5245 

8514 

5393 

8421 

5539 

8326 

5683 

8228 

22 

39 

5098 

8603 

5247 

8513 

5395 

8420 

5541 

8324 

5686 

8226 

21 

40 

5100 

8601 

5250 

8511 

5398 

8418 

5544 

8323 

5688 

8225 

20 

41 

5103 

8600 

5252 

8510 

5400 

8417 

5546 

8321 

5690 

8223 

19 

42 

5105 

8599 

5255 

8508 

5402 

8415 

5548 

8320 

5693 

8221 

18 

43 

5108 

8597 

5257 

8507 

5405 

8414 

5551 

8318 

5695 

8220 

17 

44 

5110 

8596 

5260 

8505 

5407 

8412 

5553 

8316 

5698 

8218 

16 

45 

5113 

8594 

5262 

8504 

5410 

8410 

5556 

8315 

5700 

8216 

15 

46 

5115 

8593 

5265 

8502 

5412 

8409 

5558 

8313 

5702 

8215 

14 

47 

5118 

8591 

5267 

8500 

5415 

8407 

5561 

8311 

5705 

8213 

13 

48 

5120 

8590 

5270 

8499 

5417 

8406 

5563 

8310 

5707 

8211 

12 

49 

5123 

8588 

5272 

8497 

5420 

8404 

5565 

8308 

5710 

8210 

11 

50 

5125 

8587 

5275 

8496 

5422 

8403 

5568 

8307 

5712 

8208 

lO 

51 

5128 

8585 

5277 

8494 

5424 

8401 

5570 

8305 

5714 

8207 

9 

52 

5130 

8584 

5279 

8493 

5427 

8399 

5573 

8303 

5717 

8205 

8 

53 

5133 

8582 

5282 

8491 

5429 

8398 

5575 

8302 

5719 

8203 

7 

54 

5135 

8581 

5284 

8490 

5432 

8396 

5577 

8300 

5721 

8202 

6 

55 

5138 

8579 

5287 

8488 

5434 

8395 

5580 

8299 

5724 

8200 

5 

56 

5140 

8578 

5289 

8487 

5437 

8393 

5582 

8297 

5726 

8198 

4 

57 

5143 

8576 

5292 

8485 

5439 

8391 

5585 

8295 

5729 

8197 

3 

58 

5145 

8575 

5294 

8484 

5442 

8390 

5587 

8294 

5731 

8195 

2 

59 

5148 

8573 

5297 

8482 

5444 

8388 

5590 

8292 

5733 

8193 

1 

60 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

5736 

8192 

O 

cos  sin 
59° 

cos  sin 
58° 

cos  sin 

57° 

cos  sin 
56° 

cos  sin 
55° 

f 

f 

) 

NATURAL 

SINES  AND 

COSINES. 

59 

/ 

35° 

36° 

37° 

38° 

39° 

f 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

5736 

8192 

5878 

8090 

6018 

7986 

6157 

7880 

6293 

7771 

60 

1 

5738 

8190 

5880 

8088 

6020 

7985 

6159 

7878 

6295 

7770 

59 

2 

5741 

8188 

5883 

8087 

6023 

7983 

6161 

7877 

6298 

7768 

58 

3 

5743 

8187 

5885 

8085 

6025 

7981 

6163 

7875 

6300 

7766 

57 

4 

5745 

8185 

5887 

8083 

6027 

7979 

6166 

7873 

6302 

7764 

56 

5 

5748 

8183 

5890 

8082 

6030 

7978 

6168 

7871 

6305 

7762 

55 

6 

5750 

8181 

5892 

8080 

6032 

7976 

6170 

7869 

6307 

7760 

54 

7 

5752 

8180 

5894 

8078 

6034 

7974 

6173 

7868 

6309 

7759 

53 

8 

5755 

8178  . 

5897 

8076 

6037 

7972 

6175 

7866 

6311 

7757 

52 

9 

5757 

8176 

5899 

8075 

6039 

7971 

6177 

7864 

6314 

7755 

51 

lO 

5760 

8175 

5901 

8073 

6041 

7969 

6180 

7862 

6316 

7753 

50 

11 

5762 

8173 

5904 

8071 

6044 

7967 

6182 

7860 

6318 

7751 

49 

12 

5764 

8171 

5906 

8070 

6046 

7965 

6184 

7859 

6320 

7749 

48 

13 

5767 

8170 

5908 

8068 

6048 

7964 

6186 

7857 

6323 

7748 

47 

14 

5769 

8168 

5911 

8066 

6051 

7962 

6189 

7855 

6325 

7746 

46 

15 

5771 

8166 

5913 

8064 

6053 

7960 

6191 

7853 

6327 

7744 

45 

16 

5774 

8165 

5915 

8063 

6055 

7958 

6193 

7851 

6329 

7742 

44 

17 

5776 

8163 

5918 

8061 

6058 

7956 

6196 

7850 

6332 

7740 

43 

18 

5779 

8161 

5920 

8059 

6060 

7955 

6198 

7848 

6334 

7738 

42 

19 

5781 

8160 

5922 

8058 

6062 

7953 

6200 

7346 

6336 

7737 

41 

20 

5783 

8158 

5925 

8056 

6065 

7951 

6202 

7844 

6338 

7735 

40 

21 

5786 

8156 

5927 

8054 

6067 

7950 

6205 

7842 

6341 

7733 

39 

22 

5788 

8155 

5930 

8052 

6069 

7948 

6207 

7841 

6343 

7731 

38 

23 

5790 

8153 

5932 

8051 

6071 

7946 

6209 

7839 

6345 

7729 

37 

24 

5793 

8151 

5934 

8049 

6074 

7944 

6211 

7837 

6347 

7727 

36 

25 

5795 

8150 

5937 

8047 

6076 

7942 

6214 

7835 

6350 

7725 

35 

26 

5798 

8148 

5939 

8045 

6078 

7941 

6216 

7833 

6352 

7724 

34 

27 

5800 

8146 

5941 

8044 

6081 

7939 

6218 

7832 

6354 

7722 

33 

28 

5802 

8145 

5944 

8042 

6083 

7937 

6221 

7830 

6356 

7720 

32 

29 

5805 

8143 

5946 

8040 

6085 

7935 

6223 

7828 

6359 

7718 

31 

30 

5807 

8141 

5948 

8039 

6088 

7934 

6225 

7826 

6361 

7716 

30 

31 

5809 

8139 

5951 

8037 

6090 

7932 

6227 

7824 

6363 

7714 

29 

32 

5812 

8138 

5953 

8035 

6092 

7930 

6230 

7822 

6365 

7713 

28 

33 

5814 

8136 

5955 

8033 

6095 

7928 

6232 

7821 

6368 

7711 

27 

34 

5816 

8134 

5958 

8032 

6097 

7926 

6234 

7819 

6370 

7709 

26 

35 

5819 

8133 

5960 

8030 

6099 

7925 

6237 

7817 

6372 

7707 

25 

36 

5821 

8131 

5962 

8028 

6101 

7923 

6239 

7815 

6374 

7705 

24 

37 

5824 

8129 

5965 

8026 

6104 

7921 

6241 

7813 

9376 

7703 

23 

38 

5826 

8128 

5967 

8025 

6106 

7919 

6243 

7812 

6379 

7701 

22 

39 

5828 

8126 

5969 

8023 

6108 

7918 

6246 

7810 

6381 

7700 

21 

40 

5831 

8124 

5972 

8021 

6111 

7916 

6248 

7808 

6383 

7698 

20 

41 

5833 

8123 

5974 

8020 

6113 

7914 

6250 

7806 

6385 

7696 

19 

42 

5835 

8121 

5976 

8018 

6115 

7912 

6252 

7804 

6388 

7694 

18 

43 

5838 

8119 

5979 

8016 

6118 

7910 

6255 

7802 

6390 

7692 

17 

44 

5840 

8117 

5981 

8014 

6120 

7909 

6257 

7801 

6392 

7690 

16 

45 

5842 

8116 

5983 

8013 

6122 

7907 

6259 

7799 

6394 

7688 

15 

46 

5845 

8114 

5986 

8011 

6124 

7905 

6262 

7797 

6397 

7687 

14 

47 

5847 

8112 

5988 

8009 

6127 

7903 

6264 

7795 

6399 

7685 

13 

48 

5850 

8111 

5990 

8007 

6129 

7902 

6266 

7793 

6401 

7683 

12 

49 

5852 

8109 

5993 

8006 

6131 

7900 

6268 

7792 

6403 

7681 

11 

50 

5854 

8107 

5995 

8004 

6134 

7898 

6271 

7790 

6406 

7679 

lO 

51 

5857 

8106 

5997 

8002 

6136 

7896 

6273 

7788 

6408 

7677 

9 

52 

5859 

8104 

6000 

8000 

6138 

7894 

6275 

7786 

6410 

7675 

8 

53 

5861 

8102 

6002 

7999 

6141 

7893 

6277 

7784 

6412 

7674 

7 

54 

5864 

8100 

6004 

7997 

6143 

7891 

6280 

7782 

6414 

7672 

6 

55 

5866 

8099 

6007 

7995 

6145 

7889 

6282 

7781 

6417 

7670 

5 

56 

5868 

8097 

6009 

7993 

6147 

7887 

6284 

7779 

6419 

7668 

4 

57 

5871 

8095 

6011 

7992 

6150 

7885 

6286 

7777 

6421 

7666 

3 

58 

5873 

8094 

6014 

7990 

6152 

7884 

6289 

7775 

6423 

7664 

2 

59 

5875 

8092 

6016 

7988 

6154 

7882 

6291 

7773 

6426 

7662 

1 

60 

5878 

8090 

6018 

7986 

6157 

7880 

6293 

7771 

6428 

7660 

0 

cos  sin 
54° 

cos  sin 
53° 

cos  sin 

52° 

cos  sin 
51° 

cos  sin 
50° 

f 

r 

60 

NATURAL 

SINES  AND 

COSINES. 

/ 

40° 

41° 

42° 

43° 

sin  cos 

44° 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

O 

6428 

7660 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

60 

1 

6430 

7^j59 

6563 

7545 

6693 

7430 

6822 

7312 

6949 

7191 

59 

2 

6432 

7657 

6565 

7543 

6696 

7428 

6824 

7310 

6951 

7189 

58 

3 

6435 

7655 

6567 

7541 

6698 

7426 

6826 

7308 

6953 

7187 

57 

4 

6437 

7653 

6569 

7539 

6700 

7424 

6828 

7306 

6955 

7185 

56 

5 

6439 

7651 

6572 

7538 

6702 

7422 

6831 

7304 

6957 

7183 

55 

6 

6441 

7649 

6574 

7536 

6704 

7420 

6833 

7302 

6959 

7181 

54 

7 

6443 

7647 

6576 

7534 

6706 

7418 

6835 

7300 

6961 

7179 

53 

8 

6446 

7645 

6578 

7532 

6709 

7416 

6837 

7298 

6963 

7177 

52 

9 

6448 

7644 

6580 

7530 

6711 

7414 

6839 

7296 

6965 

7175 

51 

10 

6450 

7642 

6583 

7528 

6713 

7412 

6841 

7294 

6967 

7173 

50 

11 

6452 

7640 

6585 

7526 

6715 

7410 

6843 

7292 

6970 

7171 

49 

12 

6455 

7638 

6587 

7524 

6717 

7408 

6845 

7290 

6972 

7169 

48 

13 

6457 

7636 

6589 

7522 

6719 

7406 

6848 

7288 

6974 

7167 

47 

14 

6459 

7634 

6591 

7520 

6722 

7404 

6850 

7286 

6976 

7165 

46 

16 

6461 

7632 

6593 

7518 

6724 

7402 

6852 

7284 

6978 

7163 

45 

16 

6463 

7630 

6596 

7516 

6726 

7400 

6854 

7282 

6980 

7161 

44 

17 

6466 

7629 

6598 

7515 

6728 

7398 

6856 

7280 

6982 

7159 

43 

18 

6468 

7627 

6600 

7513 

6730 

7396 

6858 

7278 

6984 

7157 

42 

19 

6470 

7625 

6602 

7511 

6732 

7394 

6860 

7276 

6986 

7155 

41 

20 

6472 

7623 

6604 

7509 

6734 

7392 

6862 

7274 

6988 

7153 

40 

21 

6475 

7621 

6607 

7507 

6737 

7390 

6865 

7272 

6990 

7151 

39 

22 

6477 

7619 

6609 

7505 

6739 

7388 

6867 

7270 

6992 

7149 

38 

23 

6479 

7617 

6611 

7503 

6741 

7387 

6869 

7268 

6995 

7147 

37 

24 

6481 

7615 

6613 

7501 

6743 

7385 

6871 

7266 

6997 

7145 

36 

25 

6483 

7613 

6615 

7499 

6745 

7383 

6873 

7264 

6999 

7143 

35 

26 

6486 

7612 

6617 

7497 

6747 

7381 

6875 

7262 

7001 

7141 

34 

27 

6488 

7610 

6620 

7495 

6749 

7379 

6877 

7260 

7003 

7139 

33 

28 

6490 

7608 

6622 

7493 

6752 

7377 

6879 

7258 

7005 

7137 

31 

29 

6492 

7606 

6624 

7491 

6754 

7375 

6881 

7256 

7007 

7135 

31 

30 

6494 

7604 

6626 

7490 

6756 

7373 

6884 

7254 

7009 

7133 

30 

31 

6497 

7602 

6628 

7488 

6758 

7371 

6886 

7252 

7011 

7130 

29 

32 

6499 

7600 

6631 

7486 

6760 

7369 

6888 

7250 

7013 

7128 

28 

33 

6501 

7598 

6633 

7484 

6762 

7367 

6890 

7248 

7015 

7126 

27 

34 

6503 

7596 

6635 

7482 

6764 

7365 

6892 

7246 

7017 

7124 

26 

35 

6506 

7595 

6637 

7480 

6767 

7363 

6894 

7244 

7019 

7122 

25 

36 

6508 

7593 

6639 

7478 

6769 

7361 

6896 

7242 

7022 

7120 

24 

37 

6510 

7591 

6641 

7476 

6771 

7359 

6898 

7240 

7024 

7118 

23 

38 

6512 

7589 

6644 

7474 

6773 

7357 

6900 

7238 

7026 

7116 

22 

39 

6514 

7587 

6646 

7472 

6775 

7355 

6903 

7236 

7028 

7114 

21 

40 

6517 

7585 

6648 

7470 

6777 

7253 

6905 

7234 

7030 

7112 

20 

41 

6519 

7583 

6650 

7468 

6779 

7351 

6907 

7232 

7032 

7110 

19 

42 

6521 

7581 

6652 

7466 

6782 

7349 

6909 

7230 

7034 

7108 

18 

43 

6523 

7579 

6654 

7464 

6784 

7347 

6911 

7228 

7036 

7106 

17 

44 

6525 

7578 

6657 

7463 

6786 

7345 

6913 

7226 

7038 

7104 

16 

45 

6528 

7576 

6659 

7461 

6788 

7343 

6915 

7224 

7040 

7102 

15 

46 

6530 

7574 

6661 

7459 

6790 

7341 

6917 

7222 

7042 

7100 

14 

47 

6532 

7572 

6663 

7457 

6792 

7339 

6919 

7220 

7044 

7098 

13 

48 

6534 

7570 

6665 

7455 

6794 

7337 

6921 

7218 

7046 

7096 

12 

49 

6536 

7568 

6667 

7453 

6797 

7335 

6924 

7216 

7048 

7094 

11 

50 

6539 

7566 

6670 

7451 

6799 

7333 

6926 

7214 

7050 

7092 

lO 

51 

6541 

7564 

6672 

7449 

6801 

7331 

6928 

7212 

7053 

7090 

9 

52 

6543 

7562 

6674 

7447 

6803 

7329 

6930 

7210 

7055 

7088 

8 

53 

6545 

7560 

6676 

7445 

6805 

7327 

6932 

7208 

7057 

7085 

7 

54 

6547 

7559 

6678 

7443 

6807 

7325 

6934 

7206 

7059 

7083 

6 

55 

6550 

7557 

6680 

7441 

6809 

7323 

6936 

7203 

7061 

7081 

5 

56 

6552 

7555 

6683 

7439 

6811 

7321 

6938 

7201 

7063 

7079 

4 

57 

6554 

7553 

6685 

7437 

6814 

7319 

6940 

7199 

7065 

7077 

3 

58 

6556 

7551 

6687 

7435 

6816 

7318 

6942 

7197 

7067 

7075 

2 

59 

6558 

7549 

6689 

7433 

6818 

7316 

6944 

7195 

7069 

7073 

1 

60 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

7071 

7071 

O 

cos   sin 
49° 

cos  sin 
48° 

cos 

sin 

cos  sin 
46° 

cos   sin 

45° 

f 

47° 

f 

NAT 

UKAl 

^  TANGENTS  AND  COTANGENTS. 

61 

f 

o° 

1° 

2° 

3° 

4° 

f 

tan 

cot 

tan 

cot 

tan   cot 

tan 

cot    tan 

cot 

o 

0000 

Infinite 

0175 

57.2900 

0349  28.6363 

0524 

19.0811  0699 

14.3007 

60 

1 

0003 

3437.75 

0177 

56.3506 

0352  28.3994 

0527 

18.9755  0702 

14.2411 

59 

2 

0006 

1718.87 

0180 

55.4415 

0355  28.1664 

0530 

18.8711  0705 

14.1821 

58 

3 

0009 

1145.92 

0183 

54.5613 

0358  27.9372 

0533 

18.7678  0708 

14.1235 

57 

4 

0012 

859.436 

0186 

53.7086 

0361  27.7117 

0536 

18.6656  0711 

14.0655 

56 

5 

0015 

687.549 

0189 

52.8821 

0364  27.4899 

0539 

18.5645  0714 

14.0079 

55 

6 

0017 

572.957 

0192 

52.0807 

0367  27.2715 

0542 

18.4645  0717 

13.9507 

54 

7 

0020 

491.106 

0195 

51.3032 

0370  27.0566 

0544 

18.3655  0720 

13.8940 

53 

8 

0023 

429.718 

0198 

50.5485 

0373  26.8450 

0547 

18.2677  0723 

13.8378 

52 

9 

0026 

381.971 

0201 

49.8157 

0375  26.6367 

0550 

18.1708  0726 

13.7821 

51 

10 

0029 

343.774 

0204 

49.1039 

0378  26.4316 

0553 

18.0750  0729 

13.7267 

50 

11 

0032 

312.521 

0207 

48.4121 

0381  26.2296 

0556 

17.9802  0731 

13.6719 

49 

12 

0035 

286.478 

0209 

47.7395 

0384  26.0307 

0559 

17.8863  0734 

13.6174 

48 

13 

0038 

264.441 

0212 

47.0853 

0387  25.8348 

0562 

17.7934  0737 

13.5634 

47 

14 

0041 

245.552 

0215 

46.4489 

0390  25.6418 

0565 

17.7015  0740 

13.5098 

46 

15 

0044 

229.182 

0218 

45.8294 

0393  25.4517 

0568 

17.6106  0743 

13.4566 

45 

16 

0047 

214.858 

0221 

45.2261 

0396  25.2644 

0571 

17.5205  0746 

13.4039 

44 

17 

0049 

202.219 

0224 

44.6386 

0399  25.0798 

0574 

17.4314  0749 

13.3515 

43 

18 

0052 

190.984 

0227 

44.0661 

0402  24.8978 

0577 

17.3432  0752 

13.2996 

42 

19 

0055 

180.932 

0230 

43.5081 

0405  24.7185 

0580 

17.2558  0755 

13.2480 

41 

20 

0058 

171.885 

0233 

42.9641 

0407  24.5418 

0582 

17.1693  0758 

13.1969 

40 

21 

0061 

163.700 

0236 

42.4335 

0410  24.3675 

0585 

17.0837  0761 

13.1461 

39 

22 

0064 

156.259 

0239 

41.9158 

0413  24.1957 

0588 

16.9990  0764 

13.0958 

38 

23 

0067 

149.465 

0241 

41.4106 

0416  24.0263 

0591 

16.9150  0767 

13.0458 

37 

24 

0070 

143.237 

0244 

40.9174 

0419  23.8593 

0594 

16.8319  0769 

12.9962 

36 

25 

0073 

137.507 

0247 

40.4358 

0422  23.6945 

0597 

16.7496  0772 

12.9469 

35 

26 

0076 

132.219 

0250 

39.9655 

0425  23.5321 

0600 

16.6681  0775 

12.8981 

34 

27 

0079 

127.321 

0253 

39.5059 

0428  23.3718 

0603 

16.5874  0778 

12.8496 

33 

28 

0081 

122.774 

0256 

39.0568 

0431  23.2137 

0606 

16.5075  0781 

12.8014 

32 

29 

0084 

118.540 

0259 

38.6177 

0434  23.0577 

0609 

16.4283  0784 

12.7536 

31 

30 

0087 

114.589 

0262 

38.1885 

0437  22.9038 

0612 

16.3499  0787 

12.7062 

30 

31 

0090 

110.892 

0265 

37.7686 

0440  22.7519 

0615 

16.2722  0790 

12.6591 

29 

32 

0093 

107.426 

0268 

37.3579 

0442  22.6020 

0617 

16.1952  0793 

12.6124 

28 

ZZ 

0096 

104.171 

0271 

36.9560 

0445  22.4541 

0620 

16.1190  0796 

12.5660 

27 

34 

0099 

101.107 

0274 

36.5627 

0448  22.3081 

0623 

16.0435  0799 

12.5199 

26 

35 

0102 

98.2179 

0276 

36.1776 

0451  22.1640 

0626 

15.9687  0802 

12.4742 

25 

36 

0105 

95.4895 

0279 

35.8006 

0454  22.0217 

0629 

15.8945  0805 

12.4288 

24 

37 

0108 

92.9085 

0282 

35.4313 

0457  21.8813 

0632 

15.8211  0808 

12.3838 

23 

38 

0111 

90.4633 

0285 

35.0695 

0460  21.7426 

0635 

15.7483  0810 

12.3390 

22 

39 

0113 

88.1436 

0288 

34.7151 

0463  21.6056 

0638 

15.6762  0813 

12.2946 

21 

40 

0116 

85.9398 

0291 

34.3678 

0466  21.4704 

0641 

15.6048  0816 

12.2505 

20 

41 

0119 

83.8435 

0294 

34.0273 

0469  21.3369 

0644 

15.5340  0819 

12.2067 

19 

42 

0122 

81.8470 

0297 

33.6935 

0472  21.2049 

0647 

15.4638  0822 

12.1632 

18 

43 

0125 

79.9434 

0300 

33.3662 

0475  21.0747 

0650 

15.3943  0825 

12.1201 

17 

44 

0128 

78.1263 

0303 

33.0452 

0477  20.9460 

0653 

15.3254  0828 

12.0772 

16 

45 

0131 

76.3900 

0306 

32.7303 

0480  20.8188 

0655 

15.2571  0831 

12.0346 

15 

46 

0134 

74.7292 

0308 

32.4213 

0483  20.6932 

0658 

15.1893  0834 

11.9923 

14 

47 

0137 

73.1390 

0311 

32.1181 

0486  20.5691 

0661 

15.1222  0837 

11.9504 

13 

48 

0140 

71.6151 

0314 

31.8205 

0489  20.4465 

0664 

15.0557  0840 

11.9087 

12 

49 

0143 

70.1533 

0317 

31.5284 

0492  20.3253 

0667 

14.9898  0843 

11.8673 

11 

50 

0146 

68.7501 

0320 

31.2416 

0495  20.2056 

0670 

14.9244  0846 

11.8262 

lO 

51 

0148 

67.4019 

0323 

30.9599 

0498  20.0872 

0673 

14.8596  0849 

11.7853 

9 

52 

0151 

66.1055 

0326 

30.6833 

0501  19.9702 

0676 

14.7954  0851 

11.7448 

8 

53 

0154 

64.8580 

0329 

30.4116 

0504  19.8546 

0679 

14.7317  0854 

11.7045 

7 

54 

0157 

63.6567 

0332 

30.1446 

0507  19.7403 

0682 

14.6685  0857 

11.6645 

6 

55 

0160 

62.4992 

0335 

29.8823 

0509  19.6273 

0685 

14.6059  0860 

11.6248 

5 

56 

0163 

61.3829 

0338 

29.6245 

0512  19.5156 

0688 

14.5438  0863 

11.5853 

4 

57 

0166 

60.3058 

0340 

29.3711 

0515  19.4051 

0690 

14.4823  0866 

11.5461 

3 

58 

0169 

59.2659 

0343 

29.1220 

0518  19.2959 

0693 

14.4212  0869 

11.5072 

2 

59 

0172 

58.2612 

0346 

28.8771 

0521  19.1879 

0696 

14.3607  0872 

11.4685 

1 

60 

0175 

57.2900 

0349 

28.6363 

0524  19.0811 

0699 

14.3007  0875 

11.4301 

O 

cot 

tan 

cot 

tan 

cot   tan 

cot 

tan    cot 

tan 

f 

89° 

88° 

87° 

86°       85° 

/ 

62 

NATURAL  TANGENTS  AND 

COTANGENTS. 

r 

5° 

6^ 

70 

8° 

9° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

0875 

11.4301 

1051 

9.5144 

1228 

8.1443 

1405 

7.1154 

1584 

6.3138 

60 

1 

0878 

11.3919 

1054 

9.4878 

1231 

8.1248 

1408 

7.1004 

1587 

6.3019 

59 

2 

0881 

11.3540 

1057 

9.4614 

1234 

8.1054 

1411 

7.0855 

1590 

6.2901 

58 

3 

0884 

11.3163 

1060 

9.4352 

1237 

8.0860 

1414 

7.0706 

1593 

6.2783 

57 

4 

0887 

11.2789 

1063 

9.4090 

1240 

8.0667 

1417 

7.0558 

1596 

6.2666 

56 

5 

0890 

11.2417 

1066 

9.3831 

1243 

8.0476 

1420 

7.0410 

1599 

6.2549 

55 

6 

0892 

11.2048 

1069 

9.3572 

1246 

8.0285 

1423 

7.0264 

1602 

6.2432 

54 

7 

0895 

11.1681 

1072 

9.3315 

1249 

8.0095 

1426 

7.0117 

1605 

6.2316 

53 

8 

0898 

11.1316 

1075 

9.3060 

1251 

7.9906 

1429 

6.9972 

1608 

6.2200 

52 

9 

0901 

11.0954 

1078 

9.2806 

1254 

7.9718 

1432 

6.9827 

1611 

6.2085 

51 

lO 

0904 

11.0594 

1080 

9.2553 

1257 

7.9530 

1435 

6.9682 

1614 

6.1970 

50 

11 

0907 

11.0237 

1083 

9.2302 

1260 

7.9344 

1438 

6.9538 

1617 

6.1856 

49 

12 

0910 

10.9882 

1086 

9.2052 

1263 

7.9158 

1441 

6.9395 

1620 

6.1742 

48 

13 

0913 

10.9529 

1089 

9.1803 

1266 

7.8973 

1444 

6.9252 

1623 

6.1628 

47 

14 

0916 

10.9178 

1092 

9.1555 

1269 

7.8789 

1447 

6.9110 

1626 

6.1515 

46 

15 

0919 

10.8829 

1095 

9.1309 

1272 

7.8606 

1450 

6.8969 

1629 

6.1402 

45 

16 

0922 

10.8483 

1098 

9.1065 

1275 

7.8424 

1453 

6.8828 

1632 

6.1290 

44 

17 

0925 

10.8139 

1101 

9.0821 

1278 

7.8243 

1456 

6.8687 

1635 

6.1178 

43 

18 

0928 

10.7797 

1104 

9.0579 

1281 

7.8062 

1459 

6.8548 

1638 

6.1066 

42 

19 

0931 

10.7457 

1107 

9.0338 

1284 

7.7883 

1462 

6.8408 

1641 

6.0955 

41 

20 

0934 

10.7119 

1110 

9.0098 

1287 

7.7704 

1465 

6.8269 

1644 

6.0844 

40 

21 

0936 

10.6783 

1113 

8.9860 

1290 

7.7525 

1468 

6.8131 

1647 

6.0734 

39 

22 

0939 

10.6450 

1116 

8.9623 

1293 

7.7348 

1471 

6.7994 

1650 

6.0624 

38 

23 

0942 

10.6118 

1119 

8.9387 

1296 

7.7171 

1474 

6.7856 

1653 

6.0514 

37 

24 

0945 

10.5789 

1122 

8.9152 

1299 

7.6996 

1477 

6.7720 

1655 

6.0405 

36 

25 

0948 

10.5462 

1125 

8.8919 

1302 

7.6821 

1480 

6.7584 

1658 

6.0296 

35 

26 

0951 

10.5136 

1128 

8.8686 

1305 

7.6647 

1483 

6.7448 

1661 

6.0188 

34 

27 

0954 

10.4813 

1131 

8.8455 

1308 

7.6473 

1486 

6.7313 

1664 

6.0080 

33 

28 

0957 

10.4491 

1134 

8.8225 

1311 

7.6301 

1489 

6.7179 

1667 

5.9972 

32 

29 

0960 

10.4172 

1136 

8.7996 

1314 

7.6129 

1492 

6.7045 

1670 

5.9865 

31 

30 

0963 

10.3854 

1139 

8.7769 

1317 

7.5958 

1495 

6.6912 

1673 

5.9758 

30 

31 

0966 

10.3538 

1142 

8.7542 

1319 

7.5787 

1497 

6.6779 

1676 

5.9651 

29 

32 

0969 

10.3224 

1145 

8.7317 

1322 

7.5618 

1500 

6.6646 

1679 

5.9545 

28 

33 

0972 

10.2913 

1148 

8.7093 

1325 

7.5449 

1503 

6.6514 

1682 

5.9439 

27 

34 

0975 

10.2602 

1151 

8.6870 

1328 

7.5281 

1506 

6.6383 

1685 

5.9333 

26 

35 

0978 

10.2294 

1154 

8.6648 

1331 

7.5113 

1509 

6.6252 

1688 

5.9228 

25 

36 

0981 

10.1988 

1157 

8.6427 

1334 

7.4947 

1512 

6.6122 

1691 

5.9124 

24 

37 

0983 

10.1683 

1160 

8.6208 

1337 

7.4781 

1515 

6.5992 

1694 

5.9019 

23 

38 

0986 

10.1381 

1163 

8.5989 

1340 

7.4615 

1518 

6.5863 

1697 

5.8915 

22 

39 

0989 

10.1080 

1166 

8.5772 

1343 

7.4451 

1521 

6.5734 

1700 

5.8811 

21 

40 

0992 

10.0780 

1169 

8.5555 

1346 

7.4287 

1524 

6.5606 

1703 

5.8708 

20 

41 

0995 

10.0483 

1172 

8.5340 

1349 

7.4124 

1527 

6.5478 

1706 

5.8605 

19 

42 

0998 

10.0187 

1175 

8.5126 

1352 

7.3962 

1530 

6.5350 

1709 

5.8502 

18 

43 

1001 

9.9893 

1178 

8.4913 

1355 

7.3800 

1533 

6.5223 

1712 

5.8400 

17 

44 

1004 

9.9601 

1181 

8.4701 

1358 

7.3639 

1536 

6.5097 

1715 

5.8298 

16 

45 

1007 

9.9310 

1184 

8.4490 

1361 

7.3479 

1539 

6.4971 

1718 

5.8197 

15 

46 

1010 

9.9021 

1187 

8.4280 

1364 

7.3319 

1542 

6.4846 

1721 

5.8095 

14 

47 

1013 

9.8734 

1189 

8.4071 

1367 

7.3160 

1545 

6.4721 

1724 

5.7994 

13 

48 

1016 

9.8448 

1192 

8.3863 

1370 

7.3002 

1548 

6.4596 

1727 

5.7894 

12 

49 

1019 

9.8164 

1195 

8.3656  • 

1373 

7.2844 

1551 

6.4472 

1730 

5.7794 

11 

50 

1022 

9.7882 

1198 

8.3450 

1376 

7.2687 

1554 

6.4348 

1733 

5.7694 

10 

51 

1025 

9.7601 

1201 

8.3245 

1379 

7.2531 

1557 

6.4225 

1736 

5.7594 

9 

52 

1028 

9.7322 

1204 

8.3041 

1382 

7.2375 

1560 

6.4103 

1739 

5.7495 

8 

53 

1030 

9.7044 

1207 

8.2838 

1385 

7.2220 

1563 

6.3980 

1742 

5.7396 

7 

54 

1033 

9.6768 

1210 

8.2636 

1388 

7.2066 

1566 

6.3859 

1745 

5.7297 

6 

55 

1036 

9.6499 

1213 

8.2434 

1391 

7.1912 

1569 

6.3737 

1748 

5.7199 

5 

56 

1039 

96220 

1216 

8.2234 

1394 

7.1759 

1572 

6.3617 

1751 

5.7101 

4 

57 

1042 

9.5949 

1219 

8.2035 

1397 

7.1607 

1575 

6.3496 

1754 

5.7004 

3 

58 

1045 

9.5679 

1222 

8.1837 

1399 

7.1455 

1578 

6.3376 

1757 

5.6906 

2 

59 

1048 

9.5411 

1225 

8.1640 

1402 

7.1304 

1581 

6.3257 

1760 

5.6809 

1 

60 

1051 

9.5144 

1228 

8.1443 

1405 

7.1154 

1584 

6.3138 

1763 

5.6713 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

r 

84° 

83° 

82° 

81° 

80° 

/ 

NATURAL  TANGENTS  AND 

COTANGENTS. 

63 

f 

lO^ 

11^ 

12° 

13° 

14° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

1763 

5.6713 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

60 

1 

1766 

5.6617 

1947 

5.1366 

2129 

4.6979 

2312 

4.3257 

2496 

4.0058 

59 

2 

1769 

5.6521 

3950 

5.1286 

2132 

4.6912 

2315 

4.3200 

2499 

4.0009 

58 

3 

1772 

5.6425 

1953 

5.1207 

2135 

4.6845 

2318 

4.3143 

2503 

3.9959 

57 

4 

1775 

5.6330 

1956 

5.1128 

2138 

4.6779 

2321 

4.3086 

2506 

3.9910 

56 

6 

1778 

5.6234 

1-959 

5.1049 

2141 

4.6712 

2324 

4.3029 

2509 

3.9861 

55 

6 

1781 

5.6140 

1962 

5.0970 

2144 

4.6646 

2327 

4.2972 

2512 

3.9812 

54 

7 

1784 

5.6045 

1965 

5.0892 

2147 

4.6580 

2330 

4.2916 

2515 

3.9763 

53 

8 

1787 

5.5951 

1968 

5.0814 

2150 

4.6514 

2333 

4.2859 

2518 

3.9714 

52 

9 

1790 

5.5857 

1971 

5.0736 

2153 

4.6448 

2336 

4.2803 

2521 

3.9665 

51 

lO 

1793 

5.5764 

1974 

5.0658 

2156 

4.6382 

2339 

4.2747 

2524 

3.9617 

50 

11 

1796 

5.5671 

1977 

5.0581 

2159 

4.6317 

2342 

4.2691 

2527 

3.9568 

49 

12 

1799 

5.5578 

1980 

5.0504 

2162 

4.6252 

2345 

4.2635 

2530 

3.9520 

48 

13 

1802 

5.5485 

1983 

5.0427 

2165 

4.6187 

2349 

4.2580 

2533 

3.9471 

47 

14 

1805 

5.5393 

1986 

5.0350 

2168 

4.6122 

2352 

4.2524 

2537 

3.9423 

46 

15 

1808 

5.5301 

1989 

5.0273 

2171 

4.6057 

2355 

4.2468 

2540 

3.9375 

45 

16 

1811 

5.5209 

1992 

5.0197 

2174 

4.5993 

2358 

4.2413 

2543 

3.9327 

44 

17 

1814 

5.5118 

1995 

5.0121 

2177 

4.5928 

2361 

4.2358 

2546 

3.9279 

43 

18 

1817 

5.5026 

1998 

5.0045 

2180 

4.5864 

2364 

4.2303 

2549 

3.9232 

42 

19 

1820 

5.4936 

2001 

4.9969 

2183 

4.5800 

2367 

4.2248 

2552 

3.9184 

41 

20 

1823 

5.4845 

2004 

4.9894 

2186 

4.5736 

2370 

4.2193 

2555 

3.9136 

40 

21 

1826 

5.4755 

2007 

4.9819 

2189 

4.5673 

2373 

4.2139 

2558 

3.9089 

39 

22 

1829 

5.4665 

2010 

4.9744 

2193 

4.5609 

2376 

4.2084 

2561 

3.9042 

38 

23 

1832 

5.4575 

2013 

4.9669 

2196 

4.5546 

2379 

4.2030 

2564 

3.8995 

37 

24 

1835 

5.4486 

2016 

4.9594 

2199 

4.5483 

2382 

4.1976 

2568 

3.8947 

36 

25 

1838 

5.4397 

2019 

4.9520 

2202 

4.5420 

2385 

4.1922 

2571 

3.8900 

35 

26 

1841 

5.4308 

2022 

4.9446 

2205 

4.5357 

2388 

4.1868 

2574 

3.8854 

34 

27 

1844 

5.4219 

2025 

4.9372 

2208 

4.5294 

2392 

4.1814 

2577 

3.8807 

33 

28 

1847 

5.4131 

2028 

4.9298 

2211 

4.5232 

2395 

4.1760 

2580 

3.8760 

32 

29 

1850 

5.4043 

2031 

4.9225 

2214 

4.5169 

2398 

4.1706 

2583 

3.8714 

31 

30 

1853 

5.3955 

2035 

4.9152 

2217 

4.5107 

2401 

4.1653 

2586 

3.8667 

30 

31 

1856 

5.3868 

2038 

4.9078 

2220 

4.5045 

2404 

4.1600 

2589 

3.8621 

29 

32 

1859 

5.3781 

2941 

4.9006 

2223 

4.4983 

2407 

4.1547 

2592 

3.8575 

28 

33 

1862 

5.3694 

2044 

4.8933 

2226 

4.4922 

2410 

4.1493 

2595 

3.8528 

27 

34 

1865 

5.3607 

2047 

4.8860 

2229 

4.4860 

2413 

4.1441 

2599 

3.8482 

26 

35 

1868 

5.3521 

2050 

4.8788 

2232 

4.4799 

2416 

4.1388 

2602 

3.8436 

25 

36 

1871 

5.3435 

2053 

4.8716 

2235 

4.4737 

2419 

4.1335 

2605 

3.8391 

24 

37 

1874 

5.3349 

2056 

4.8644 

2238 

4.4676 

2422 

4.1282 

2608 

3.8345 

23 

38 

1877 

5.3263 

2059 

4.8573 

2241 

4.4615 

2425 

4.1230 

2611 

3.8299 

22 

39 

1880 

5.3178 

2062 

4.8501 

2244 

4.4555 

2428 

4.1178 

2614 

3.8254 

21 

40 

1883 

5.3093 

2065 

4.8430 

2247 

4.4494 

2432 

4.1126 

2617 

3.8208 

20 

41 

1887 

5.3008 

2068 

4.8359 

2251 

4.4434 

2435 

4.1074 

2620 

3.8163 

19 

42 

1890 

5.2924 

2071 

4.8288 

2254 

4.4374 

2438 

4.1022 

2623 

3.8118 

18 

43 

1893 

5.2839 

2074 

4.8218 

2257 

4.4313 

2441 

4.0970 

2627 

3.8073 

17 

44 

1896 

5.2755 

2077 

4.8147 

2260 

4.4253 

2444 

4.0918 

2630 

3.8028 

16 

45 

1899 

5.2672 

2080 

4.8077 

2263 

4.4194 

2447 

4.0867 

2633 

3.7983 

15 

46 

1902 

5.2588 

2083 

4.8007 

2266 

4.4134 

2450 

4.0815 

2636 

3.7938 

14 

47 

1905 

5.2505 

2086 

4.7937 

2269 

4.4075 

2453 

4.0764 

2639 

3.7893 

13 

48 

1908 

5.2422 

2089 

4.7867 

2272 

4.4015 

2456 

4.0713 

2642 

3.7848 

12 

49 

1911 

5.2339 

2092 

4.7798 

2275 

4.3956 

2459 

4.0662 

2645 

3.7804 

11 

50 

1914 

5.2257 

2095 

4.7729 

2278 

4.3897 

2462 

4.0611 

2648 

3.7760 

10 

51 

1917 

5.2174 

2098 

4.7659 

2281 

4.3838 

2465 

4.0560 

2651 

3.7715 

9 

52 

1920 

5.2092 

2101 

4.7591 

2284 

4.3779 

2469 

4.0509 

2655 

3.7671 

8 

53 

1923 

5.2011 

2104 

4.7522 

2287 

4.3721 

2472 

4.0459 

2658 

3.7627 

7 

54 

1926 

5.1929 

2107 

4.7453 

2290 

4.3662 

2475 

4.0408 

2661 

3.7583 

6 

55 

1929 

5.1848 

2110 

4.7385 

2293 

4.3604 

2478 

4.0358 

2664 

3.7539 

5 

56 

1932 

5.1767 

2113 

4.7317 

2296 

4.3546 

2481 

4.0308 

2667 

3.7495 

4 

57 

1935 

5.1686 

2116 

4.7249 

2299 

4.3488 

2484 

4.0257 

2670 

3.7451 

3 

58 

1938 

5.1606 

2119 

4.7181 

2303 

4.3430 

2487 

4.0207 

2673 

3.7408 

2 

59 

1941 

5.1^26 

2123 

4.7114 

2306 

4.3372 

2490 

4.0158 

2676 

3.7364 

1 

60 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

2679 

3.7321 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

790 

78° 

770 

76° 

75° 

/ 

64 

NATURAL  TANGENTS  AND 

COTANGENTS. 

f 

15^ 

16° 

17° 

18° 

19° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

2679 

3.7321 

2867 

3.4874 

3057 

3.2709 

3249 

3.0777 

3443 

2.9042 

60 

1 

2683 

3.7277 

2871 

3.4836 

3060 

3.2675 

3252 

3.0746 

3447 

2.9015 

59 

2 

2686 

3.7234 

2874 

3.4798 

3064 

3.2641 

3256 

3.0716 

3450 

2.8987 

58 

3 

2689 

3.7191 

2877 

3.4760 

3067 

3.2607 

3259 

3.0686 

3453 

2.8960 

57 

4 

2692 

3.7148 

2880 

3.4722 

3070 

3.2573 

3262 

3.0655 

3456 

2.8933 

56 

5 

2695 

3.7105 

2883 

3.4684 

3073 

3.2539 

3265 

3.0625 

3460 

2.8905 

55 

6 

2698 

3.7062 

2886 

3.4646 

3076 

3.2506 

3269 

3.0595 

3463 

2.8878 

54 

7 

2701 

3.7019 

2890 

3.4608 

3080 

3.2472 

3272 

3.0565 

3466 

2.8851 

53 

8 

2704 

3.6976 

2893 

3.4570 

3083 

3.2438 

3275 

3.0535 

3469 

2.8824 

52 

9 

2708 

3.6933 

2896 

3.4533 

3086 

3.2405 

3278 

3.0505 

3473 

2.8797 

51 

10 

2711 

3.6891 

2899 

3.4495 

3089 

3.2371 

3281 

3.0475 

3476 

2.8770 

50 

11 

2714 

3.6848 

2902 

3.4458 

3092 

3.2338 

3285 

3.0445 

3479 

2.8743 

49 

12 

2717 

3.6806 

2905 

3.4420 

3096 

3.2305 

3288 

3.0415 

3482 

2.8716 

48 

13 

2720 

3.6764 

2908 

3.4383 

3099 

3.2272 

3291 

3.0385 

3486 

2.8689 

47 

14 

2723 

3.6722 

2912 

3.4346 

3102 

3.2238 

3294 

3.0356 

3489 

2.8662 

46 

15 

2726 

3.6680 

2915 

3.4308 

3105 

3.2205 

3298 

3.0326. 

3492 

2.8636 

45 

16 

2729 

3.6638 

2918 

3.4271 

3108 

3.2172 

3301 

3.0296 

3495 

2.8609 

44 

17 

2733 

3.6596 

2921 

3.4234 

3111 

3.2139 

3304 

3.0267 

3499 

2.8582 

43 

18 

2736 

3.6554 

2924 

3.4197 

3115 

3.2106 

3307 

3.0237 

3502 

2.8556 

42 

19 

2739 

3.6512 

2927 

3.4160 

3118 

3.2073 

3310 

3.0208 

3505 

2.8529 

41 

20 

2742 

3.6470 

2931 

3.4124 

3121 

3.2041 

3314 

3.0178 

3508 

2.8502 

40 

21 

2745 

3.6429 

2934 

3.4087 

3124 

3.2008 

3317 

3.0149 

3512 

2.8476 

39 

22 

2748 

3.6387 

2937 

3.4050 

3127 

3.1975 

3320 

3.0120 

3515 

2.8449 

38 

23 

2751 

3.6346 

2940 

3.4014 

3131 

3.1943 

3323 

3.0090 

3518 

2.8423 

37 

24 

2754 

3.6305 

2943 

3.3977 

3134 

3.1910 

3327 

3.0061 

3522 

2.8397 

36 

25 

2758 

3.6264 

2946 

3.3941 

3137 

3.1878 

3330 

3.0032 

3525 

2.8370 

35 

26 

2761 

3.6222 

2949 

3.3904 

3140 

3.1845 

3ZZZ 

3.0003 

3528 

2.8344 

34 

27 

2764 

3.6181 

2953 

3.3868 

3143 

3.1813 

3336 

2.9974 

3531 

2.8318 

33 

28 

2767 

3.6140 

2956 

3.3832 

3147 

3.1780 

3339 

2.9945 

3535 

2.8291 

32 

29 

2770 

3.6100 

2959 

3.3796 

3150 

3.1748 

3343 

29916 

3538 

2.8265 

31 

30 

2773 

3.6059 

2962 

3.3759 

3153 

3.1716 

3346 

2.9887 

3541 

2.8239 

30 

31 

2776 

3.6018 

2965 

3.3723 

3156 

3.1684 

3349 

2.9858 

3544 

2.8213 

29 

32 

2780 

3.5978 

2968 

3.3687 

3159 

3.1652 

3352 

2.9829 

3548 

2.8187 

28 

33 

2783 

3.5937 

2972 

3.3652 

3163 

3.1620 

3356 

2.9800 

3551 

2.8161 

27 

34 

2786 

3.5897 

2975 

3.3616 

3166 

3.1588 

3359 

2.9772 

3554 

2.8135 

26 

35 

2789 

3.5856 

2978 

3.3580 

3169 

3.1556 

3362 

2.9743 

3558 

2.8109 

25 

36 

2792 

3.5816 

2981 

3.3544 

3172 

3.1524 

3365 

2.9714 

3561 

2.8083 

24 

37 

2795 

3.5776 

2984 

3.3509 

3175 

3.1492 

3369 

2.9686 

3564 

2.8057 

23 

38 

2798 

3.5736 

2987 

3.3473 

3179 

3.1460 

3372 

2.9657 

3567 

2.8032 

22 

39 

2801 

3.5696 

2991 

3.3438 

3182 

3.1429 

3375 

2.9629 

3571 

2.8006 

21 

40 

2805 

3.5656 

2994 

3.3402 

3185 

3.1397 

3378 

2.9600 

3574 

2.7980 

20 

41 

2808 

3.5616 

2997 

3.3367 

3188 

3.1366 

3382 

2.9572 

3577 

2.7955 

19 

42 

2811 

3.5576 

3000 

3.3332 

3191 

3.1334 

3385 

2.9544 

3581 

2.7929 

18 

43 

2814 

3.5536 

3003 

3.3297 

3195 

3.1303 

3388 

2.9515 

3584 

2.7903 

17 

44 

2817 

3.5497 

3006 

3.3261 

3198 

3.1271 

3391 

2.9487 

3587 

2.7878 

16 

45 

2820 

3.5457 

3010 

3.3226 

3201 

3.1240 

3395 

2.9459 

3590 

2.7852 

15 

46 

2823 

3.5418 

3013 

3.3191 

3204 

3.1209 

3398 

2.9431 

3594 

2.7827 

14 

47 

2827 

3.5379 

3016 

3.3156 

3207 

3.1178 

3401 

2.9403 

3597 

2.7801 

13 

48 

2830 

3.5339 

3019 

3.3122 

3211 

3.1146 

3404 

2.9375 

3600 

2.7776 

12 

49 

2833 

3.5300 

3022 

3.3087 

3214 

3.1115 

3408 

2.9347 

3604 

2.7751 

11 

50 

2836 

3.5261 

3026 

3.3052 

3217 

3.1084 

3411 

2.9319 

3607 

2.7725 

lO 

51 

2839 

3.5222 

3029 

3.3017 

3220 

3.1053 

3414 

2.9291 

3610 

2.7700 

9 

52 

2842 

3.5183 

3032 

3.2983 

3223 

3.1022 

3417 

2.9263 

3613 

2.7675 

8 

53 

2845 

3.5144 

3035 

3.2948 

3227 

3.0991 

3421 

2.9235 

3617 

2.7650 

7 

54 

2849 

3.5105 

3038 

3.2914 

3230 

3.0961 

3424 

2.9208 

3620 

2.7625 

6 

55 

2852 

3.5067 

3041 

3.2880 

3233 

3.0930 

3427 

2.9180 

3623 

2.7600 

5 

56 

2855 

3.5028 

3045 

3.2845 

3236 

3.0899 

3430 

2.9152 

3627 

2.7575 

4 

57 

2858 

3.4989 

3048 

3.2811 

3240 

3.0868 

3434 

2.9125 

3630 

2.7550 

3 

58 

2861 

3.4951 

3051 

3.2777 

3243 

3.0838 

3437 

2.9097 

3633 

2.7525 

2 

59 

2864 

3.4912 

3054 

3.2743 

3246 

3.0807 

3440 

2.9070 

3636 

2.7500 

1 

60 

2867 

3.4874 

3057 

3.2709 

3249 

3.0777 

3443 

2.9042 

3640 

2.7475 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

74° 

73° 

72° 

71° 

70° 

f 

NATURAL  TANGENTS  AND 

COTANGENTS. 

65 

f 

20^ 

21° 

22° 

23° 

24° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

3640 

2.7475 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

60 

1 

3643 

2.7450 

3842 

2.6028 

4044 

2.4730 

4248 

2.3539 

4456 

2.2443 

59 

2 

3646 

2.7425 

3845 

2.6006 

4047 

2.4709 

4252 

2.3520 

4459 

2.2425 

58 

3 

3650 

2.7400 

3849 

2.5983 

4050 

2.4689 

4255 

2.3501 

4463 

2.2408 

57 

4 

3653 

2.7376 

3852 

2.5961 

4054 

2.4668 

4258 

2.3483 

4466 

2.2390 

56 

5 

3656 

2.7351 

3855 

2.5938 

4057 

2.4648 

4262 

2.3464 

4470 

22373 

55 

6 

3659 

2.7326 

3859 

2.5916 

4061 

2.4627 

4265 

2.3445 

4473 

2.2355 

54 

7 

3663 

2.7302 

3862 

2.5893 

4064 

2.4606 

4269 

2.3426 

4477 

2.2338 

53 

8 

3666 

2.7277 

3865 

2.5871 

4067 

2.4586 

4272 

2.3407 

4480 

2.2320 

52 

9 

3669 

2.7253 

3869 

2.5848 

4071 

2.4566 

4276 

2.3388 

4484 

2.2303 

51 

lO 

3673 

2.7228 

3872 

2.5826 

4074 

2.4545 

4279 

2.3369 

4487 

2.2286 

50 

11 

3676 

2.7204 

3875 

2.5804 

4078 

2.4525 

4283 

2.3351 

4491 

2.2268 

49 

12 

3679 

2.7179 

3879 

2.5782 

4081 

2.4504 

4286 

2.3332 

4494 

2.2251 

48 

13 

3683 

2.7155 

3882 

2.5759 

4084 

2.4484 

4289 

2.3313 

4498 

2.2234 

47 

14 

3686 

2.7130 

3885 

2.5737 

4088 

2.4464 

4293 

2.3294 

4501 

2.2216 

46 

15 

3689 

2.7106 

3889 

2.5715 

4091 

2.4443 

4296 

2.3276 

4505 

2.2199 

45 

16 

3693 

2.7082 

3892 

2.5693 

4095 

2.4423 

4300 

2.3257 

4508 

2.2182 

44 

17 

3696 

2.7058 

3895 

2.5671 

4098 

2.4403 

4303 

2.3238 

4512 

2.2165 

43 

18 

3699 

2.7034 

3899 

2.5649 

4101 

2.4383 

4307 

2.3220 

4515 

2.2148 

42 

19 

3702 

2.7009 

3902 

2.5627 

4105 

2.4362 

4310 

2.3201 

4519 

2.2130 

41 

20 

3706 

2.6985 

3906 

2.5605 

4108 

2.4342 

4314 

2.3183 

4522 

2.2113 

40 

21 

3709 

2.6961 

3909 

2.5533 

4111 

2.4322 

4317 

2.3164 

4526 

2.2096 

39 

22 

3712 

2.6937 

3912 

2.5561 

4115 

2.4302 

4320 

2.3146 

4529 

2.2079 

38 

23 

3716 

2.6913 

3916 

2.5539 

4118 

2.4282 

4324 

2.3127 

4533 

2.2062 

37 

24 

3719 

2.6889 

3919 

2.5517 

4122 

2.4262 

4327 

2.3109 

4536 

2.2045 

36 

25 

3722 

2.6865 

3922 

2.5495 

4125 

2.4242 

4331 

2.3090 

4540 

2.2028 

35 

26 

3726 

2.6841 

3926 

2.5473 

4129 

2.4222 

4334 

2.3072 

4543 

2.2011 

34 

27 

3729 

2.6818 

3929 

2.5452 

4132 

2.4202 

4338 

2.3053 

4547 

2.1994 

33 

28 

3732 

2.6794 

3932 

2.5430 

4135 

2.4182 

4341 

2.3035 

4550 

2.1977 

31 

29 

3736 

2.6770 

3936 

2.5408 

4139 

2.4162 

4345 

2.3017 

4554 

2.1960 

31 

30 

3739 

2.6746 

3939 

2.5386 

4142 

2.4142 

4348 

2.2998 

4557 

2.1943 

30 

31 

3742 

2.6723 

3942 

2.5365 

4146 

2.4122 

4352 

2.2980 

4561 

2.1926 

29 

32 

3745 

2.6699 

3946 

2.5343 

4149 

2.4102 

4355 

2.2962 

4564 

2.1909 

28 

33 

3749 

2.6675 

3949 

2.5322 

4152 

2.4083 

4359 

2.2944 

4568 

2.1892 

27 

34 

3752 

2.6652 

3953 

2.5300 

4156 

2.4063 

4362 

2.2925 

4571 

2.1876 

26 

35 

3755 

2.6628 

3956 

2.5279 

4159 

2.4043 

4365 

2.2907 

4575 

2.1859 

25 

36 

3759 

2.6605 

3959 

2.5257 

4163 

2.4023 

4369 

2.2889 

4578 

2.1842 

24 

37 

3762 

2.6581 

3963 

2.5236 

4166 

2.4004 

4372 

2.2871 

4582 

2.1825 

23 

38 

3765 

2.6558 

3966 

2.5214 

4169 

2.3984 

4376 

2.2853 

4585 

2.1808 

22 

39 

3769 

2.6534 

3969 

2.5193 

4173 

2.3964 

4379 

2.2835 

4589 

2.1792 

21 

40 

3772 

2.6511 

3973 

2.5172 

4176 

2.3945 

4383 

2.2817 

4592 

2.1775 

20 

41 

3775 

2.6488 

3976 

2.5150 

4180 

2.3925 

4386 

2.2799 

4596 

2.1758 

19 

42 

3779 

2.6464 

3979 

2.5129 

4183 

2.3906 

4390 

2.2781 

4599 

2.1742 

18 

43 

3782 

2.6441 

3983 

2.5108 

4187 

2.3886 

4393 

2.2763 

4603 

2.1725 

17 

44 

3785 

2.6418 

3986 

2.5086 

4190 

2.3867 

4397 

2.2745 

4607 

2.1708 

16 

45 

3789 

2.6395 

3990 

2.5065 

4193 

2.3847 

4400 

2.2727 

4610 

2.1692 

15 

46 

3792 

2.6371 

3993 

2.5044 

4197 

2.3828 

4404 

2.2709 

4614 

2.1675 

14 

47 

3795 

2.6348 

3996 

2.5023 

4200 

2.3808 

4407 

2.2691 

4617 

2.1659 

13 

48 

3799 

2.6325 

4000 

2.5002 

4204 

2.3789 

4411 

2.2673 

4621 

2.1642 

12 

49 

3802 

2.6302 

4003 

2.4981 

4207 

2.3770 

4414 

2.2655 

4624 

2.1625 

11 

50 

3805 

2.6279 

4006 

2.4960 

4210 

2.3750 

4417 

2.2637 

4628 

2.1609 

10 

51 

3809 

2.6256 

4010 

2.4939 

4214 

2.3731 

4421 

2.2620 

4631 

2.1592 

9 

52 

3812 

2.6233 

4013 

2.4918 

4217 

2.3712 

4424 

2.2602 

4635 

2.1576 

8 

53 

3815 

2.6210 

4017 

2.4897 

4221 

2.3693 

4428 

2.2584 

4638 

2.1560 

7 

54 

3819 

2.6187 

4020 

2.4876 

4224 

2.3673 

4431 

2.2566 

4642 

2.1543 

6 

55 

3822 

2.6165 

4023 

2.4855 

4228 

2.3654 

4435 

2.2549 

4645 

2.1527 

5 

56 

3825 

2.6142 

4027 

2.4834 

4231 

2.3635 

4438 

2.2531 

4649 

2.1510 

4 

57 

3829 

2.6119 

4030 

2.4813 

4234 

2.3616 

4442 

2.2513 

4652 

2.1494 

3 

58 

3832 

2.6096 

4033 

2.4792 

4238 

2.3597 

4445 

2.2496 

4656 

2.1478 

2 

59 

3835 

2.6074 

4037 

2.4772 

4241 

2.3578 

4449 

2.2478 

4660 

2.1461 

1 

60 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

4663 

2.1445 

O 

f 

cot   tan 
69° 

cot   tan 
68° 

cot   tan 
67° 

cot   tan 
66° 

cot  tan 
65° 

f 

66  NATURAL  TANGENTS   AND   COTANGENTS. 


f 

25^ 

26° 

27° 

28° 

29° 

f 

tan   cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

4663  2.1445 

4877 

2.0503 

5095 

1.9626 

5317 

1.8807 

5543 

1.8040 

60 

1 

4667  2.1429 

4881 

2.0488 

5099 

1.9612 

5321 

1.8794 

5547 

1.8028 

59 

2 

4670  2.1413 

4885 

2.0473 

5103 

1.9598 

5325 

1.8781 

5551 

1.8016 

58 

3 

4674  2.1396 

4888 

2.0458 

5106 

1.9584 

5328 

1.8768 

5555 

1.8003 

57 

4 

4677  2.1380 

4892 

2.0443 

5110 

1.9570 

5332 

1.8755 

5558 

1.7991 

56 

5 

4681  2.1364 

4895 

2.0428 

5114 

1.9556 

5336 

1.8741 

5562 

1.7979 

55 

6 

4684  2.1348 

4899 

2.0413 

5117 

1.9542 

5340 

1.8728 

5566 

1.7966 

54 

7 

4688  2.1332 

4903 

2.0398 

5121 

1.9528 

'5343 

1.8715 

5570 

1.7954 

53 

8 

4691  2.1315 

4906 

2.0383 

5125 

1.9514 

5347 

1.8702 

5574 

1.7942 

52 

9 

4695  2.1299 

4910 

2.0368 

5128 

1.9500 

5351 

1.8689 

5577 

1.7930 

51 

lO 

4699  2.1283 

4913 

2.0353 

5132 

1.9486 

5354 

1.8676 

5581 

1.7917 

50 

11 

4702  2.1267 

4917 

2.0338 

5136 

1.9472 

5358 

1.8663 

5585 

1.7905 

49 

12 

4706  2.1251 

4921 

2.0323 

5139 

1.9458 

5362 

1.8650 

5589 

1.7893 

48 

13 

4709  2.1235 

4924 

2.0308 

5143 

1.9444 

5366 

1.8637 

5593 

1.7881 

47 

14 

4713  2.1219 

4928 

2.0293 

5147 

1.9430 

5369 

1.8624 

5596 

1.7868 

46 

15 

4716  2.1203 

4931 

2.0278 

5150 

1.9416 

5373 

1.8611 

5600 

1.7856 

45 

16 

4720  2.1187 

4935 

2.0263 

5154 

1.9402 

5377 

1.8598 

5604 

1.7844 

44 

17 

4723  2.1171 

4939 

2.0248 

5158 

1.9388 

5381 

1.8585 

5608 

1.7832 

43 

18 

4727  2.1155 

4942 

2.0233 

5161 

1.9375 

5384 

1.8572 

5612 

1.7820 

42 

19 

4731  2.1139 

4946 

2.0219 

5165 

1.9361 

5388 

1.8559 

5616 

1.7808 

41 

20 

4734  2.1123 

4950 

2.0204 

5169 

1.9347 

5392 

1.8546 

5619 

1.7796 

40 

21 

4738  2.1107 

4953 

2.0189 

5172 

1.9333 

5396 

1.8533 

5623 

1.7783 

39 

22 

4741  2.1092 

4957 

2.0174 

5176 

1.9319 

5399 

1.8520 

5627 

1.7771 

38 

23 

4745  2.1076 

4960 

2.0160 

5180 

1.9306 

5403 

1.8507 

5631 

1.7759 

37 

24 

4748  2.1060 

4964 

2.0145 

5184 

1.9292 

5407 

1.8495 

5635 

1.7747 

36 

25 

4752  2.1044 

4968 

2.0130 

5187 

1.9278 

5411 

1.8482 

5639 

1.7735 

35 

26 

4755  2.1028 

4971 

2.0115 

5191 

1.9265 

5415 

1.8469 

5642 

1.7723 

34 

27 

4759  2.1013 

4975 

2.0101 

5195 

1.9251 

5418 

1.8456 

5646 

1.7711 

33 

28 

4763  2.0997 

4979 

2.0086 

5198 

1.9237 

5422 

1.8443 

5650 

1.7699 

31 

29 

4766  2.0981 

4982 

2.0072 

5202 

1.9223 

5426 

1.8430 

5654 

1.7687 

31 

30 

4770  2.0965 

4986 

2.0057 

5206 

1.9210 

5430 

1.8418 

5658 

1.7675 

30 

31 

4773  2.0950 

4989 

2.0042 

5209 

1.9196 

5433 

1.8405 

5662 

1.7663 

29 

32 

4777  2.0934 

4993 

2.0028 

5213 

1.9183 

5437 

1.8392 

5665 

1.7651 

28 

33 

4780  2.0918 

4997 

2.0013 

5217 

1.9169 

5441 

1.8379 

5669 

1.7639 

27 

34 

4784  2.0903 

5000 

1.9999 

5220 

1.9155 

5445 

1.8367 

5673 

1.7627 

26 

35 

4788  2.0887 

5004 

1.9984 

5224 

1.9142 

5448 

1.8354 

5677 

1.7615 

25 

36 

4791  2.0872 

5008 

1.9970 

5228 

1.9128 

5452 

1.8341 

5681 

1.7603 

24 

37 

4795  2.0856 

5011 

1.9955 

5232 

1.9115 

5456 

1.8329 

5685 

1.7591 

23 

38 

4798  2.0840 

5015 

1.9941 

5235 

1.9101 

5460 

1.8316 

5688 

1.7579 

22 

39 

4802  2.0825 

5019 

1.9926 

5239 

1.9088 

5464 

1.8303 

5692 

1.7567 

21 

40 

4806  2.0809 

5022 

1.9912 

5243 

1.9074 

5467 

1.8291 

5696 

1.7556 

20 

41 

4809  2.0794 

5026 

1.9897 

5246 

1.9061 

5471 

1.8278 

5700 

1.7544 

19 

42 

4813  2.0778 

5029 

1.9883 

5250 

1.9047 

5475 

1.8265 

5704 

1.7532 

18 

43 

4816  2.0763 

5033 

1.9868 

5254 

1.9034 

5479 

1.8253 

5708 

1.7520 

17 

44 

4820  2.0748 

5037 

1.9854 

5258 

1.9020 

5482 

1.8240 

5712 

1.7508 

16 

45 

4823  2.0732 

5040 

1.9840 

5261 

1.9007 

5486 

1.8228 

5715 

1.7496 

15 

46 

4827  2.0717 

5044 

1.9825 

5265 

1.8993 

5490 

1.8215 

5719 

1.7485 

14 

47 

4831  2.0701 

5048 

1.9811 

5269 

1.8980 

5494 

1.8202 

5723 

1.7473 

13 

48 

4834  2.0686 

5051 

1.9797 

5272 

1.8967 

5498 

1.8190 

5727 

1.7461 

12 

49 

4838  2.0671 

5055 

1.9782 

5276 

1.8953 

5501 

1.8177 

5731 

1.7449 

11 

50 

4841  2.0655 

5059 

1.9768 

5280 

1.8940 

5505 

1.8165 

5735 

1.7437 

10 

51 

4845  2.0640 

5062 

1.9754 

5284 

1.8927 

5509 

1.8152 

5739 

1.7426 

9 

52 

4849  2.0625 

5066 

1.9740 

5287 

1.8913 

5513 

1.8140 

5743 

1.7414 

8 

53 

4852  2.0609 

5070 

1.9725 

5291 

1.8900 

5517 

1.8127 

5746 

1.7402 

7 

54 

4856  2.0594 

5073 

1.9711 

5295 

1.8887 

5520 

1.8115 

5750 

1.7391 

6 

55 

4859  2.0579 

5077 

1.9697 

5298 

1.8873 

5524 

1.8103 

5754 

1.7379 

5 

56 

4863  2.0564 

5081 

1.9683 

5302 

1.8860 

5528 

1.8090 

5758 

1.7367 

4 

57 

4867  2.0549 

5084 

1.9669 

5306 

1.8847 

5532 

1.8078 

5762 

1.7355 

3 

58 

4870  2.0533 

5088 

1.9654 

5310 

1.8834 

5535 

1.8065 

5766 

1.7344 

2 

59 

4874  2.0518 

5092 

1.9640 

5313 

1.8820 

5539 

1.8053 

5770 

1.7332 

1 

60 

4877  2.0503 

5095 

1.9626 

5317 

1.8807 

5543 

1.8040 

5774 

1.7321 

O 

cot   tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

64° 

63° 

62° 

61° 

60° 

f 

NATURAL  TANGENTS  AND 

COTANGENTS. 

67 

f 

30^ 

31° 

32° 

33° 

34° 

r 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

5774 

1.7321 

6009 

1.6643 

6249 

1.6003 

6494 

1.5399 

6745 

1.4826 

60 

1 

STn 

1.7309 

6013 

1.6632 

6253 

1.5993 

6498 

1.5389 

6749 

1.4816 

59 

2 

5781 

1.7297 

6017 

1.6621 

6257 

1.5983 

6502 

1.5379 

6754 

1.4807 

58 

3 

5785 

1.7286 

6020 

1.6610 

6261 

1.5972 

6506 

1.5369 

6758 

1.4798 

57 

4 

5789 

1.7274 

6024 

1.6599 

6265 

1.5962 

6511 

1.5359 

6762 

1.4788 

56 

5 

5793 

1.7262 

6028 

1.6588 

6269 

1.5952 

6515 

1.5350 

6766 

1.4779 

55 

6 

5797 

1.7251 

6032 

1.6577 

6273 

1.5941 

6519 

1.5340 

6771 

1.4770 

54 

7 

5801 

1.7239 

6036 

1.6566 

6277 

1.5931 

6523 

1.5330 

6775 

1.4761 

53 

8 

5805 

1.7228 

6040 

1.6555 

6281 

1.5921 

6527 

1.5320 

6779 

1.4751 

52 

9 

5808 

1.7216 

60H 

1.6545 

6285 

1.5911 

6531 

1.5311 

6783 

1.4742 

51 

lO 

5812 

1.7205 

6(H8 

1.6534 

6289 

1.5900 

6536 

1.5301 

6787 

1.4733 

50 

11 

5816 

1.7193 

6052 

1.6523 

6293 

1.5890 

6540 

1.5291 

6792 

1.4724 

49 

12 

5820 

1.7182 

6056 

1.6512 

6297 

1.5880 

6544 

1.5282 

6796 

1.4715 

48 

13 

5824 

1.7170 

6060 

1.6501 

6301 

1.5869 

6548 

1.5272 

6800 

1.4705 

47 

14 

5828 

1.7159 

6064 

1.6490 

6305 

1.5859 

6552 

1.5262 

6805 

1.4696 

46 

15 

5832 

1.7147 

6068 

1.6479 

6310 

1.5849 

6556 

1.5253 

6809 

1.4687 

45 

16 

5836 

1.7136 

6072 

1.6469 

6314 

1.5839 

6560 

1.5243 

6813 

1.4678 

44 

17 

5840 

1.7124 

6076 

1.6458 

6318 

1.5829 

6565 

1.5233 

6817 

1.4669 

43 

18 

5844 

1.7113 

6080 

1.6447 

6322 

1.5818 

6569 

1.5224 

6822 

1.4659 

42 

19 

5847 

1.7102 

6084 

1.6436 

6326 

1.5808 

6573 

1.5214 

6826 

1.4650 

41 

20 

5851 

1.7090 

6088 

1.6426 

6330 

1.5798 

6577 

1.5204 

6830 

1.4641 

40 

21 

5855 

1.7079 

6092 

1.6415 

6334 

1.5788 

6581 

1.5195 

6834 

1.4632 

39 

22 

5859 

1.7067 

6096 

1.6404 

6338 

1.5778 

6585 

1.5185 

6839 

1.4623 

38 

23 

5863 

1.7056 

6100 

1.6393 

6342 

1.5768 

6590 

1.5175 

6843 

1.4614 

37 

24 

5867 

1.7045 

6104 

1.6383 

6346 

1.5757 

6594 

1.5166 

6847 

1.4605 

36 

25 

5871 

1.7033 

6108 

1.6372 

6350 

1.5747 

6598 

1.5156 

6851 

1.4596 

35 

26 

5875 

1.7022 

6112 

1.6361 

6354 

1.5737 

6602 

1.5147 

6856 

1.4586 

34 

27 

5879 

1.7011 

6116 

1.6351 

6358 

1.5727 

6606 

1.5137 

6860 

1.4577 

33 

28 

5883 

1.6999 

6120 

1.6340 

6363 

1.5717 

6610 

1.5127 

6864 

1.4568 

32 

29 

5887 

1.6988 

6124 

1.6329 

6367 

1.5707 

6615 

1.5118 

6869 

1.4559 

31 

30 

5890 

1.6977 

6128 

1.6319 

6371 

1.5697 

6619 

1.5108 

6873 

1.4550 

30 

31 

5894 

1.6965 

6132 

1.6308 

6375 

1.5687 

6623 

1.5099 

6877 

1.4541 

29 

32 

5898 

1.6954 

6136 

1.6297 

6379 

1.5677 

6627 

1.5089 

6881 

1.4532 

28 

33 

5902 

1.6943 

6140 

1.6287 

6383 

1.5667 

6631 

1.5080 

6886 

1.4523 

27 

34 

5906 

1.6932 

6144 

1.6276 

6387 

1.5657 

6636 

1.5070 

6890 

1.4514 

26 

35 

5910 

1.6920 

6148 

1.6265 

6391 

1.5647 

6640 

1.5061 

6894 

1.4505 

25 

36 

5914 

1.6909 

6152 

1.6255 

6395 

1.5637 

6644 

1.5051 

6899 

1.4496 

24 

37 

5918 

1.6898 

6156 

1.6244 

6399 

1.5627 

6648 

1.5042 

6903 

1.4487 

23 

38 

5922 

1.6887 

6160 

1.6234 

6403 

1.5617 

6652 

1.5032 

6907 

1.4478 

22 

39 

5926 

1.6875 

6164 

1.6223 

6408 

1.5607 

6657 

1.5023 

6911 

1.4469 

21 

40 

5930 

1.6864 

6168 

1.6212 

6412 

1.5597 

6661 

1.5013 

6916 

1.4460 

20 

41 

5934 

1.6853 

6172 

1.6202 

6416 

1.5587 

6665 

1.5004 

6920 

1.4451 

19 

42 

5938 

1.6842 

6176 

1.6191 

6420 

1.5577 

6669 

1.4994 

6924 

1.4442 

18 

43 

5942 

1.6831 

6180 

1.6181 

6424 

1.5567 

6673 

1.4985 

6929 

1.4433 

17 

44 

5945 

1.6820 

6184 

1.6170 

6428 

1.5557 

6678 

1.4975 

6933 

1.4424 

16 

45 

5949 

1.6808 

6188 

1.6160 

6432 

1.5547 

6682 

1.4966 

6937 

1.4415 

15 

46 

5953 

1.6797 

6192 

1.6149 

6436 

1.5537 

6686 

1.4957 

6942 

1.4406 

14 

47 

5957 

1.6786 

6196 

1.6139 

6440 

1.5527 

6690 

1.4947 

6946 

1.4397 

13 

48 

5961 

1.6775 

6200 

1.6128 

6445 

1.5517 

6694 

1.4938 

6950 

1.4388 

12 

49 

5965 

1.6764 

6204 

1.6118 

6449 

1.5507 

6699 

1.4928 

6954 

1.4379 

11 

50 

5969 

1.6753 

6208 

1.6107 

6453 

1.5497 

6703 

1.4919 

6959 

1.4370 

lO 

51 

5973 

1.6742 

6212 

1.6097 

6457 

1.5487 

6707 

1.4910 

6963 

1.4361 

9 

52 

5977 

1.6731 

6216 

1.6087 

6461 

1.5477 

6711 

1.4900 

6967 

1.4352 

8 

53 

5981 

1.6720 

6220 

1.6076 

6465 

1.5468 

6716 

1.4891 

6972 

1.4344 

7 

54 

5985 

1.6709 

6224 

1.6066 

6469 

1.5458 

6720 

1.4882 

6976 

1.4335 

6 

55 

5989 

1.6698 

6?,?,8 

1.6055 

6473 

1.5448 

6724 

1.4872 

6980 

1.4326 

5 

56 

5993 

1.6687 

6233 

1.6045 

6478 

1.5438 

6728 

1.4863 

6985 

1.4317 

4 

57 

5997 

1.6676 

6237 

1.6034 

6482 

1.5428 

6732 

1.4854 

6989 

1.4308 

3 

58 

6001 

1.6665 

6241 

1.6024 

6486 

1.5418 

6737 

1.4844 

6993 

1.4299 

2 

59 

6005 

1.6654 

6245 

1.6014 

6490 

1.5408 

6741 

1.4835 

6998 

1.4290 

1 

60 

6009 

1.6643 

6249 

1.6003 

6494 

1.5399 

6745 

1.4826 

7002 

1.4281 

O 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

r 

59° 

58° 

57° 

56° 

55° 

t 

68 

KATURAL  TANGENTS  AND 

COTANGENTS. 

f 

35^ 

36° 

37° 

38° 

39° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

7002 

1.4281 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

60 

1 

7006 

1.4273 

7270 

1.3755 

7540 

1.3262 

7818 

1.2792 

8103 

1.2342 

59 

2 

7011 

1.4264 

7274 

1.3747 

7545 

1.3254 

7822 

1.2784 

8107 

1.2334 

58 

3 

7015 

1.4255 

7279 

1.3739 

7549 

1.3246 

7827 

1.2776 

8112 

1.2327 

57 

4 

7019 

1.4246 

7283 

1.3730 

7554 

1.3238 

7832 

1.2769 

8117 

1.2320 

56 

5 

7024 

1.4237 

7288 

1.3722 

7558 

1.3230 

7836 

1.2761 

8122 

1.2312 

55 

6 

7028 

1.4229 

7292 

1.3713 

7563 

1.3222 

7841 

1.2753 

8127 

1.2305 

54 

7 

7032 

1.4220 

7297 

1.3705 

7568 

1.3214 

7846 

1.2746 

8132 

1.2298 

53 

8 

7037 

1.4211 

7301 

1.3697 

7572 

1.3206 

7850 

1.2738 

8136 

1.2290 

52 

9 

7041 

1.4202 

7306 

1.3688 

7577 

1.3198 

7855 

1.2731 

8141 

1.2283 

51 

lO 

7046 

1.4193 

7310 

1.3680 

7581 

1.3190 

7860 

1.2723 

8146 

1.2276 

50 

11 

7050 

1.4185 

7314 

1.3672 

7586 

1.3182 

7865 

1.2715 

8151 

1.2268 

49 

12 

7054 

1.4176 

7319 

1.3663 

7590 

1.3175 

7869 

1.2708 

8156 

1.2261 

48 

13 

7059 

1.4167 

7323 

1.3655 

7595 

1.3167 

7874 

1.2700 

8161 

1.2254 

47 

14 

7063 

1.4158 

7328 

1.3647 

7600 

1.3159 

7879 

1.2693 

8165 

1.2247 

46 

15 

7067 

1.4150 

7332 

1.3638 

7604 

1.3151 

7883 

1.2685 

8170 

1.2239 

45 

16 

7072 

1.4141 

7337 

1.3630 

7609 

1.3143 

7888 

1.2677 

8175 

1.2232 

44 

17 

7076 

1.4132 

7341 

1.3622 

7613 

1.3135 

7893 

1.2670 

8180 

1.2225 

43 

18 

7080 

1.4124 

7346 

1.3613 

7618 

1.3127 

7898 

1.2662 

8185 

1.2218 

42 

19 

7085 

1.4115 

7350 

1.3605 

7623 

1.3119 

7902 

1.2655 

8190 

1.2210 

41 

20 

7089 

1.4106 

7355 

1.3597 

7627 

1.3111 

7907 

1.2647 

8195 

1.2203 

40 

21 

7094 

1.4097 

7359 

1.3588 

7632 

1.3103 

7912 

1.2640 

8199 

1.2196 

39 

22 

7098 

1.4089 

7364 

1.3580 

7636 

1.3095 

7916 

1.2632 

8204 

1.2189 

38 

23 

7102 

1.4080 

7368 

1.3572 

7641 

1.3087 

7921 

1.2624 

8209 

1.2181 

37 

24 

7107 

1.4071 

7373 

1.3564 

7646 

1.3079 

7926 

1.2617 

8214 

1.2174 

36 

25 

7111 

1.4063 

7377 

1.3555 

7650 

1.3072 

7931 

1.2609 

8219 

1.2167 

35 

26 

7115 

1.4054 

7382 

1.3547 

7655 

1.3064 

7935 

1.2602 

8224 

1.2160 

34 

27 

7120 

1.4045 

7386 

1.3539 

7659 

1.3056 

7940 

1.2594 

8229 

1.2153 

33 

28 

7124 

1.4037 

7391 

1.3531 

7664 

1.3048 

7945 

1.2587 

8234 

1.2145 

32 

29 

7129 

1.4028 

7395 

1.3522 

7669 

1.3040 

7950 

1.2579 

8238 

1.2138 

31 

30 

7133 

1.4019 

7400 

1.3514 

7673 

1.3032 

7954 

1.2572 

8243 

1.2131 

30 

31 

7137 

1.4011 

7404 

1.3506 

7678 

1.3024 

7959 

1.2564 

8248 

1.2124 

29 

32 

7142 

1.4002 

7409 

1.3498 

7683 

1.3017 

7964 

1.2557 

8253 

1.2117 

28 

33 

7146 

1.3994 

7413 

1.3490 

7687 

1.3009 

7969 

1.2549 

8258 

1.2109 

27 

34 

7151 

1.3985 

7418 

1.3481 

7692 

1.3001 

7973 

1.2542 

8263 

1.2102 

26 

35 

7155 

1.3976 

7422 

1.3473 

7696 

1.2993 

7978 

1.2534 

8268 

1.2095 

25 

36 

7159 

1.3968 

7427 

1.3465 

7701 

1.2985 

7983 

1.2527 

8273 

1.2088 

24 

37 

7164 

1.3959 

7431 

1.3457 

7706 

1.2977 

7988 

1.2519 

8278 

1.2081 

23 

38 

7168 

1.3951 

7436 

1.3449 

7710 

1.2970 

7992 

1.2512 

8283 

1.2074 

22 

39 

7173 

1.3942 

7440 

1.3440 

7715 

1.2962 

7997 

1.2504 

8287 

1.2066 

21 

40 

7177 

1.3934 

7445 

1.3432 

7720 

1.2954 

8002 

1.2497 

8292 

1.2059 

20 

41 

7181 

1.3925 

7449 

1.3424 

7724 

1.2946 

8007 

1.2489 

8297 

1.2052 

19 

42 

7186 

1.3916 

7454 

1.3416 

7729 

1.2938 

8012 

1.2482 

8302 

1.2045 

18 

43 

7190 

1.3908 

7458 

1.3408 

7734 

1.2931 

8016 

1.2475 

8307 

1.2038 

17 

44 

7195 

1.3899 

7463 

1.3400 

7738 

1.2923 

8021 

1.2467 

8312 

1.2031 

16 

45 

7199 

1.3891 

7467 

1.3392 

7743 

1.2915 

8026 

1.2460 

8317 

1.2024 

15 

46 

7203 

1.3882 

7472 

1.3384 

7747 

1.2907 

8031 

1.2452 

8322 

1.2017 

14 

47 

7208 

1.3874 

7476 

1.3375 

7752 

1.2900 

8035 

1.2445 

8327 

1.2009 

13 

48 

7212 

1.3865 

7481 

1.3367 

7757 

1.2892 

8040 

1.2437 

8332 

1.2002 

12 

49 

7217 

1.3857 

7485 

1.3359 

7761 

1.2884 

8045 

1.2430 

8337 

1.1995 

11 

50 

7221 

1.3848 

7490 

1.3351 

7766 

1.2876 

8050 

1.2423 

8342 

1.1988 

lO 

51 

7226 

1.3840 

7495 

1.3343 

7771 

1.2869 

8055 

1.2415 

8346 

1.1981 

9 

52 

7230 

1.3831 

7499 

1.3335 

7775 

1.2861 

8059 

1.2408 

8351 

1.1974 

8 

53 

7234 

1.3823 

7504 

1.3327 

7780 

1.2853 

8064 

1.2401 

8356 

1.1967 

7 

54 

7S39 

1.3814 

7508 

1.3319 

7785 

1.2846 

8069 

1.2393 

8361 

1.1960 

6 

55 

7243 

1.3806 

7513 

1.3311 

7789 

1.2838 

8074 

1.2386 

8366 

1.1953 

5 

56 

7248 

1.3798 

7517 

1.3303 

7794 

1.2830 

8079 

1.2378 

8371 

1.1946 

4 

57 

.7252 

1.3789 

7522 

1.3295 

7799 

1.2822 

8083 

1.2371 

8376 

1.1939 

3 

58 

7257 

1.3781 

7526 

1.3287 

7803 

1.2815 

8088 

1.2364 

8381 

1.1932 

2 

59 

7261 

1.3772 

7531 

1.3278 

7808 

1.2807 

8093 

1.2356 

8386 

1.1925 

1 

60 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

8391 

1.1918 

O 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

54° 

53° 

52° 

51° 

50° 

f 

N. 

(^TUR. 

IL  TANGENTS  AND 

COTANGENTS. 

69 

r 

40^ 

41° 

42° 

43° 

44° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan   cot 

O 

8391 

1.1918 

8693 

1.1504 

9004 

1.1106 

9325 

1.0724 

9657  1.0355 

60 

1 

8396 

1.1910 

8698 

1.1497 

9009 

1.1100 

9331 

1.0717 

9663  1.0349 

59 

2 

8401 

1.1903 

8703 

1.1490 

9015 

1.1093 

9336 

1.0711 

9668  1.0343 

58 

3 

8406 

1.1896 

8708 

1.1483 

9020 

1.1087 

9341 

1.0705 

9674  1.0337 

57 

4 

8411 

1.1889 

8713 

1.1477 

9025 

1.1080 

9347 

1.0699 

9679  1.0331 

56 

5 

8416 

1.1882 

8718 

1.1470 

9030 

1.1074 

9352 

1.0692 

9685  1.0325 

55 

6 

8421 

1.1875 

8724 

1.1463 

9036 

1.1067 

9358 

1.0686 

9691  1.0319 

54 

7 

8426 

1.1868 

8729 

1.1456 

9041 

1.1061 

9363 

1.0680 

9696  1.0313 

53 

8 

8431 

1.1861 

8734 

1.1450 

9046 

1.1054 

9369 

1.0674 

9702  1.0307 

52 

9 

8436 

1.1854 

8739 

1.1443 

9052 

1.1048 

9374 

1.0668 

9708  1.0301 

51 

10 

8441 

1.1847 

8744 

1.1436 

9057 

1.1041 

9380 

1.0661 

9713  1.0295 

50 

11 

8446 

1.1840 

8749 

1.1430 

9062 

1.1035 

9385 

1.0655 

9719  1.0289 

49 

12 

8451 

1.1833 

8754 

1.1423 

9067 

1.1028 

9391 

1.0649 

9725  1.0283 

48 

13 

8456 

1.1826 

8759 

1.1416 

9073 

1.1022 

9396 

1.0643 

9730  1.0277 

47 

14 

8461 

1.1819 

8765 

1.1410 

9078 

1.1016 

9402 

1.0637 

9736  1.0271 

46 

16 

8466 

1.1812 

8770 

1.1403 

9083 

1.1009 

9407 

1.0630 

9742  1.0265 

46 

16 

8471 

1.1806 

8775 

1.1396 

9089 

1.1003 

9413 

1.0624 

9747  1.0259 

44 

17 

8476 

1.1799 

8780 

1.1389 

9094 

1.0996 

9418 

1.0618 

9753  1.0253 

43 

18 

8481 

1.1792 

8785 

1.1383 

9099 

1.0990 

9424 

1.0612 

9759  1.0247 

42 

19 

8486 

1.1785 

8790 

1.1376 

9105 

1.0983 

9429 

1.0606 

9764  1.0241 

41 

20 

8491 

1.1778 

8796 

1.1369 

9110 

1.0977 

9435 

1.0599 

9770  1.0235 

40 

21 

8496 

1.1771 

8801 

1.1363 

9115 

1.0971 

9440 

1.0593 

9776  1.0230 

39 

22 

8501 

1.1764 

8806 

1.1356 

9121 

1.0964 

9446 

1.0587 

9781  1.0224 

38 

23 

8506 

1.1757 

8811 

1.1349 

9126 

1.0958 

9451 

1.0581 

9787  1.0218 

37 

24 

8511 

1.1750 

8816 

1.1343 

9131 

1.0951 

9457 

1.0575 

9793  1.0212 

36 

25 

8516 

1.1743 

8821 

1.1336 

9137 

1.0945 

9462 

1.0569 

9798  1.0206 

35 

26 

8521 

1.1736 

8827 

1.1329 

9142 

1.0939 

9468 

1.0562 

9804  1.0200 

34 

27 

8526 

1.1729 

8832 

1.1323 

9147 

1.0932 

9473 

1.0556 

9810  1.0194 

33 

28 

8531 

1.1722 

8837 

].1316 

9153 

1.0926 

9479 

1.0550 

9816'  1.0188 

32 

29 

8536 

1.1715 

8842 

1.1310 

9158 

1.0919 

9484 

1.0544 

9821  1.0182 

31 

30 

8541 

1.1708 

8847 

1.1303 

9163 

1.0913 

9490 

1.0538 

9827  1.0176 

30 

31 

8546 

1.1702 

8852 

1.1296 

9169 

1.0907 

9495 

1.0532 

9833  1.0170 

29 

32 

8551 

1.1695 

8858 

1.1290 

9174 

1.0900 

9501 

1.0526 

9838  1.0164 

28 

33 

8556 

1.1688 

8863 

1.1283 

9179 

1.0894 

9506 

1.0519 

9844  1.0158 

27 

34 

8561 

1.1681 

8868 

1.1276 

9185 

1.0888 

9512 

1.0513 

9850  1.0152 

26 

35 

8566 

1.1674 

8873 

1.1270 

9190 

1.0881 

9517 

1.0507 

9856  1.0147 

25 

36 

8571 

1.1667 

8878 

1.1263 

9195 

1.0875 

9523 

1.0501 

9861  1.0141 

24 

37 

8576 

1.1660 

8884 

1.1257 

9201 

1.0869 

9528 

1.0495 

9867  1.0135 

23 

38 

8581 

1.1653 

8889 

1.1250 

9206 

1.0862 

9534 

1.0489 

9873  1.0129 

22 

39 

8586 

1.1647 

8894 

1.1243 

9212 

1.0856 

9540 

1.0483 

9879  1.0123 

21 

40 

8591 

1.1640 

8899 

1.1237 

9217 

1.0850 

9545 

1.0477 

9884  1.0117 

20 

41 

8596 

1.1633 

8904 

1.1230 

9222 

1.0843 

9551 

1.0470 

9890  1.0111 

19 

42 

8601 

1.1626 

8910 

1.1224 

9228 

1.0837 

9556 

1.0464 

9896  1.0105 

18 

43 

8606 

1.1619 

8915 

1.1217 

9233 

1.0831 

9562 

1.0458 

9902  1.0099 

17 

44 

8611 

1.1612 

8920 

1.1211 

9239 

1.0824 

9567 

1.0452 

9907  1.0094 

16 

45 

8617 

1.1606 

8925 

1.1204 

9244 

1.0818 

9573 

1.0446 

9913  1.0088 

15 

46 

8622 

1.1599 

8931 

1.1197 

9249 

1.0812 

9578 

1.0440 

9919  1.0082 

14 

47 

8627 

1.1592 

8936 

1.1191 

9255 

1.0805 

9584 

1.0434 

9925  1.0076 

13 

48 

8632 

1.1585 

8941 

1.1184 

9260 

1.0799 

9590 

1.0428 

9930  1.0070 

12 

49 

8637 

1.1578 

8946 

1.1178 

9266 

1.0793 

9595 

1.0422 

9936  1.0064 

11 

50 

8642 

1.1571 

8952 

1.1171 

9271 

1.0786 

9601 

1.0416 

9942  1.0058 

lO 

51 

8647 

1.1565 

8957 

1.1165 

9276 

1.0780 

9606 

1.0410 

9948  1.0052 

9 

52 

8652 

1.1558 

8962 

1.1158 

9282 

1.0774 

9612 

1.0404 

9954  1.0047 

8 

53 

8657 

1.1551 

8967 

1.1152 

9287 

1.0768 

9618 

1.0398 

9959  1.0041 

7 

54 

8662 

1.1544 

8972 

1.1145 

9293 

1.0761 

9623 

1.0392 

9965  1.0035 

6 

55 

8667 

1.1538 

8978 

1.1139 

9298 

1.0755 

9629 

1.0385 

9971  1.0029 

5 

56 

8672 

1.1531 

8983 

1.1132 

9303 

1.0749 

9634 

1.0379 

9977  1.0023 

4 

57 

8678 

1.1524 

8988 

1.1126 

9309 

1.0742 

9640 

1.0373 

9983  1.0017 

3 

58 

8683 

1.1517 

8994 

1.1119 

9314 

1.0736 

9646 

1.0367 

9988  1.0012 

2 

59 

8688 

1.1510 

8999 

1.1113 

9320 

1.0730 

9651 

1.0361 

9994  1.0006 

1 

60 

8693 

1.1504 

9004 

1.1106 

9325 

1.0724 

9657 

1.0355 

1.000  1.0000 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot   tan 

f 

49^ 

48° 

47° 

46° 

46° 

f 

70 

TABLE  VII. 

-TRAVERSE  TAELE. 

Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

O       f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o     / 

015 

1.000 

0.004 

2.000 

0.009 

3.000 

0.013 

4.000 

0.017 

5.000 

0.022 

89  45 

30 

1.000 

0.009 

2.000 

0.017 

3.000 

0.026 

4.000 

0.035 

5.000 

0.044 

30 

45 

1.000 

0.013 

2.000 

0.026 

3.000 

0.039 

4.000 

0.052 

5.000 

0.065 

15 

1    0 

1.000 

0.017 

2.000 

0.035 

3.000 

0.052 

3.999 

0.070 

4.999 

0.087 

89  0 

15 

1.000 

0.022 

2.000 

0.044 

2.999 

0.065 

3.999 

0.087 

4.999 

0.109 

45 

30 

1.000 

0.026 

1.999 

0.052 

2.999 

0.079 

3.999 

0.105 

4.998 

0.131 

30 

45 

1.000 

0.031 

1.999 

0.061 

2.999 

0.092 

3.998 

0.122 

4.998 

0.153 

15 

2    0 

0.999 

0.035 

1.999 

0.070 

2.998 

0.105 

3.998 

0.140 

4.997 

0.174 

88   0 

15 

0.999 

0.039 

1.998 

0.079 

2.998 

0.118 

3.997 

0.157 

4.996 

0.196 

45 

30 

0.999 

0.044 

1.998 

0.087 

2.997 

0.131 

3.996 

0.174 

4.995 

0.218 

30 

45 

0.999 

0.048 

1.998 

0.096 

2.997 

0.144 

3.995 

0.192 

4.994 

0.240 

15 

3   0 

0.999 

0.052 

1.997 

0.105 

2.996 

0.157 

3.995 

0.209 

4.993 

0.262 

87   0 

15 

0.998 

0.057 

1.997 

0.113 

2.995 

0.170 

3.994 

0.227 

4.992 

0.283 

45 

30 

0.998 

0.061 

1.996 

0.122 

2.994 

0.183 

3.993 

0.244 

4.991 

0.305 

30 

45 

0.998 

0.065 

1.996 

0.131 

2.994 

0.196 

3.991 

0.262 

4.989 

0.327 

15 

4   0 

0.998 

0.070 

1.995 

0.140 

2.993 

0.209 

3.990 

0.279 

4.988 

0.349 

86   0 

15 

0.997 

0.074 

1.995 

0.148 

2.992 

0.222 

3.989 

0.296 

4.986 

0.371 

45 

30 

0.997 

0.078 

1.994 

0.157 

2.991 

0.235 

3.988 

0.314 

4.985 

0.392 

30 

45 

0.997 

0.083 

1.993 

0.166 

2.990 

0.248 

3.986 

0.331 

4.983 

0.414 

15 

5   0 

0.996 

0.087 

1.992 

0.174 

2.989 

0.261 

3.985 

0.349 

4.981 

0.436 

85   0 

15 

0.996 

0.092 

1.992 

0.183 

2.987 

0.275 

3.983 

0.366 

4.979 

0.458 

45 

30 

0.995 

0.096 

1.991 

0.192 

2.986 

0.288 

3.982 

0.383 

4.977 

0.479 

30 

45 

0.995 

0.100 

1.990 

0.200 

2.985 

0.301 

3.980 

0.401 

4.975 

0.501 

15 

6   0 

0.995 

0.105 

1.989 

0.209 

2.984 

0.314 

3.978 

0.418 

4.973 

0.523 

84   0 

15 

0.994 

0.109 

1.988 

0.218 

2.982 

0.327 

3.976 

0.435 

4.970 

0.544 

45 

30 

0.994 

0.113 

1.987 

0.226 

2.981 

0.340 

3.974 

0.453 

4.968 

0.566 

30 

45 

0.993 

0.118 

1.986 

0.235 

2.979 

0.353 

3.972 

0.470 

4.965 

0.588 

15 

7    0 

0.993 

0.122 

1.985 

0.244 

2.978 

0.366 

3.970 

0.487 

4.963 

0.609 

83   0 

15 

0.992 

0.126 

1.984 

0.252 

2.976 

0.379 

3.968 

0.505 

4.960 

0.631 

45 

30 

0.991 

0.131 

1.983 

0.261 

2.974 

0.392 

3.966 

0.522 

4.957 

0.653 

30 

45 

0.991 

0.135 

1.982 

0.270 

2.973 

0.405 

3.963 

0.539 

4.954 

0.674 

15 

8   0 

0.990 

0.139 

1.981 

0.278 

2.971 

0.418 

3.961 

0.557 

4.951 

0.696 

82   0 

15 

0.990 

0.143 

1.979 

0.287 

2.969 

0.430 

3.959 

0.574 

4.948 

0.717 

45 

30 

0.989 

0.148 

1.978 

0.296 

2.967 

0.443 

3.956 

0.591 

4.945 

0.739 

30 

45 

0.988 

0.152 

1.977 

0.3(H 

2.965 

0.456 

3.953 

0.608 

4.942 

0.761 

15 

9   0 

0.988 

0.156 

1.975 

0.313 

2.963 

0.469 

3.951 

0.626 

4.938 

0.782 

81    0 

15 

0.987 

0.161 

1.974 

0.321 

2.961 

0.482 

3.948 

0.643 

4.935 

0.804 

45 

30 

0.986 

0.165 

1.973 

0.330 

2.959 

0.495 

3.945 

0.660 

4.931 

0.825 

30 

45 

0.986 

0.169 

1.971 

0.339 

2.957 

0.508 

3.942 

0.677 

4.928 

0.847 

15 

lO   0 

0.985 

0.174 

1.970 

0.347 

2.954 

0.521 

3.939 

0.695 

4.924 

0.868 

80   0 

15 

0.984 

0.178 

1.968 

0.356 

2.952 

0.534 

3.936 

0.712 

4.920 

0890 

45 

30 

0.983 

0.182 

1.967 

0.364 

2.950 

0.547 

3.933 

0.729 

4.916 

0.911 

30 

45 

0.982 

0.187 

1.965 

0.373 

2.947 

0.560 

3.930 

0.746 

4.912 

0.933 

15 

11    0 

0.982 

0.191 

1.963 

0.382 

2.945 

0.572 

3.927 

0.763 

4.908 

0.954 

79   0 

15 

0.981 

0.195 

1.962 

0.390 

2.942 

0.585 

3.923 

0.780 

4.904 

0.975 

45 

30 

0.980 

0.199 

1.960 

0.399 

2.940 

0.598 

3.920 

0.797 

4.900 

0.997 

30 

45 

0.979 

0.204 

1.958 

0.407 

2.937 

0611 

3.916 

0.815 

4.895 

1.018 

15 

12    0 

0.978 

0.208 

1.956 

0.416 

2.934 

0.624 

3.913 

0.832 

4.891 

1.040 

78    0 

15 

0.977 

0.212 

1.954 

0.424 

2.932 

0.637 

3.909 

0.849 

4  886 

1.061 

45 

30 

0.976 

0.216 

1.953 

0.433 

2.929 

0.649 

3.905 

0.866 

4.881 

1.082 

30 

45 

0.975 

0.221 

1.951 

0.441 

2.926 

0  662 

3.901 

0883 

4.877 

1.103 

15 

13   0 

0.974 

0.225 

1.949 

0.450 

2.923 

0.675 

3.897 

0.900 

4.872 

1.125 

77    0 

15 

0.973 

0.229 

1.947 

0.458 

2.920 

0.688 

3.894 

0.917 

4.867 

1.146 

45 

30 

0.972 

0.233 

1.945 

0.467 

2.917 

0.700 

3.889 

0.934 

4.862 

1.167 

30 

45 

0.971 

0.238 

1.943 

0.475 

2.914 

0.713 

3.885 

0.951 

4.857 

1.188 

15 

14   0 

0.970 

0.242 

1.941 

0.484 

2.911 

0.726 

3.881 

0.968 

4.851 

1.210 

76   0 

15 

0.969 

0.246 

1.938 

0.492 

2.908 

0.738 

3.877 

0.985 

4.846 

1.231 

45 

30 

0.968 

0.250 

1.936 

0.501 

2.904 

0.751 

3.873 

1.002 

4.841 

1.252 

30 

45 

0.967 

0.255 

1.934 

0.509 

2.901 

0.764 

3.868 

1.018 

4.835 

1.273 

15 

16    0 

0.966 

0.259 

1.932 

0.518 

2.898 

0.776 

3.864 

1.035 

4.830 

1.294 

75   0 

o    r 

Dep. 

Lat. 

Dep.       Lat. 
Distance  2. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.     Lat. 
Distance  5. 

9     ' 

Bearing. 

Distance  1. 

Distance  3. 

Distance  4. 

Bearing. 

75°-90' 


0°- 

-16° 

71 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

o      f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o     r 

0  15 

6.000 

0.026 

7.000 

0.031 

8.000 

0.035 

9.000 

0.039 

10.000 

0.044 

89  45 

30 

6.000 

0.052 

7.000 

0.061 

8.000 

0.070 

9.000 

0.079 

10.000 

0.087 

30 

45 

5.999 

0.079 

6.999 

0.092 

7.999 

0.105 

8.999 

0.118 

9.999 

0.131 

15 

1    0 

5.999 

0.105 

6.999 

0.122 

7.999 

0.140 

8.999 

0.157 

9.999 

0.175 

89   0 

15 

5.999 

0.131 

6.998 

0.153 

7.998 

0.175 

8.998 

0.196 

9.998 

0.218 

45 

30 

5.998 

0.157 

6.998 

0.183 

7.997 

0.209 

8.997 

0.236 

9.997 

0.262 

30 

45 

5.997 

0.183 

6.997 

0.214 

7.996 

0.244 

8.996 

0.275 

9.995 

0.305 

15 

2   0 

5.996 

0.209 

6.996 

0.244 

7.995 

0.279 

8.995 

0.314 

9.994 

0.349 

88   0 

15 

5.995 

0.236 

6.995 

0.275 

7.994 

0.314 

8.993 

0.353 

9.992 

0.393 

45 

30 

5.994 

0.262 

6.993 

0.305 

7.992 

0.349 

8.991 

0.393 

9.991 

0.436 

30 

45 

5.993 

0.288 

6.992 

0.336 

7.991 

0.384 

8.990 

0.432 

9.989 

0.480 

15 

3   0 

5.992 

0.314 

6.990 

0.366 

7.989 

0.419 

8.988 

0.471 

9.986 

0.523 

87   0 

15 

5.990 

0.340 

6.989 

0.397 

7.987 

0.454 

8.986 

0.510 

9.984 

0.567 

45 

30 

5.989 

0.366 

6.987 

0.427 

7.985 

0.488 

8.983 

0.549 

9.981 

0.611 

30 

45 

5.987 

0.392 

6.985 

0.458 

7.983 

0.523 

8.981 

0.589 

9.979 

0.654 

15 

4   0 

5.985 

0.419 

6.983 

0.488 

7.981 

0.558 

8.978 

0.628 

9.976 

0.698 

86   0 

15 

5.984 

0.445 

6.981 

0.519 

7.978 

0.593 

8.975 

0.667 

9.973 

0.741 

45 

30 

5.982 

0.471 

6.978 

0.549 

7.975 

0.628 

8.972 

0.706 

9.969 

0.785 

30 

45 

5.979 

0.497 

6.976 

0.580 

7.973 

0.662 

8.969 

0.745 

9.966 

0.828 

15 

5    0 

5.977 

0.523 

6.973 

0.610 

7.970 

0.697 

8.966 

0.784 

9.962 

0.872 

85   0 

15 

5.975 

0.549 

6.971 

0.641 

7.966 

0.732 

8.962 

0.824 

9.958 

0.915 

45 

30 

5.972 

0.575 

6.968 

0.671 

7.963 

0.767 

8.959 

0.863 

9.954 

0.959 

30 

45 

5.970 

0.601 

6.965 

0.701 

7.960 

0.802 

8.955 

0.902 

9.950 

1.002 

15 

6   0 

5.967 

0.627 

6.962 

0.732 

7.956 

0.836 

8.951 

0.941 

9.945 

1.045 

84    0 

15 

5.964 

0.653 

6.958 

0.762 

7.952 

0.871 

8.947 

0.980 

9.941 

1.089 

45 

30 

5.961 

0.679 

6.955 

0.792 

7.949 

0.906 

8.942 

1.019 

9.936 

1.132 

30 

45 

5.958 

0.705 

6.951 

0.823 

7.945 

0.940 

8.938 

1.058 

9.931 

1.175 

15 

7   0 

5.955 

0.731 

6.948 

0.853 

7.940 

0.975 

8.933 

1.097 

9.926 

1.219 

83   0 

15 

5.952 

0.757 

6.944 

0.883 

7.936 

1.010 

8.928 

1.136 

9.920 

1.262 

45 

30 

5.949 

0.783 

6.940 

0.914 

7.932 

1.044 

8.923 

1.175 

9.914 

1.305 

30 

45 

5.945 

0.809 

6.936 

0.944 

7.927 

1.079 

8.918 

1.214 

9.909 

1.349 

15 

8   0 

5.942 

0.835 

6.932 

0.974 

7.922 

1.113 

8.912 

1.253 

9.903 

1.392 

82    0 

15 

5.938 

0.861 

6.928 

1.004 

7.917 

1.148 

8.907 

1.291 

9.897 

1.435 

45 

30 

5.934 

0.887 

6.923 

1.035 

7.912 

1.182 

8.901 

1.330 

9.890 

1.478 

30 

45 

5.930 

0.913 

6.919 

1.065 

7.907 

1.217 

8.895 

1.369 

9.884 

1.521 

15 

9    0 

5.926 

0.939 

6.914 

1.095 

7.902 

1.251 

8.889 

1.408 

9.877 

1.564 

81    0 

15 

5.922 

0.964 

6.909 

1.125 

7.896 

1.286 

8.883 

1.447 

9.870 

1.607 

45 

30 

5.918 

0.990 

6.904 

1.155 

7.890 

1.320 

8.877 

1.485 

9.863 

1.651 

30 

45 

5.913 

1.016 

6.899 

1.185 

7.884 

1.355 

8^70 

1.524 

9.856 

1.694 

15 

10    0 

5.909 

1.042 

6.894 

1.216 

7.878 

1.389 

8.863 

1.563 

9.848 

1.737 

80   0 

15 

5.904 

1.068 

6.888 

1.246 

7.872 

1.424 

8.856 

1.601 

9.840 

1.779 

45 

30 

5.900 

1.093 

6.883 

1.276 

7.866 

1.458 

8.849 

1.640 

9.833 

1.822 

30 

45 

5.895 

1.119 

6.877 

1.306 

7.860 

1.492 

8.842 

1.679 

9.825 

1.865 

15 

11    0 

5.890 

1.145 

6.871 

1.336 

7.853 

1.526 

8.835 

1.717 

9.816 

1.908 

79   0 

15 

5.885 

1.171 

6.866 

1.366 

7.846 

1.561 

8.827 

1.756 

9.808 

1.951 

45 

30 

5.880 

1.196 

6.859 

1.396 

7.839 

1.595 

8.819 

1.794 

9.799 

1.994 

30 

45 

5.874 

1.222 

6.853 

1.425 

7.832 

1.629 

8.811 

1.833 

9.791 

2.036 

15 

12   0 

5.869 

1.247 

6.847 

1.455 

7.825 

1.663 

8.803 

1.871 

9.782 

2.079 

78    0 

15 

5.863 

1.273 

6.841 

1.485 

7.818 

1.697 

8.795 

1.910 

9.772 

2.122 

45 

30 

5.858 

1.299 

6.834 

1.515 

7.810 

1.732 

8.787 

1.948 

9.763 

2.164 

30 

45 

5.852 

1.324 

6.827 

1.545 

7.803 

1.766 

8.778 

1.986 

9.753 

2.207 

15 

13   0 

5.846 

1.350 

6.821 

1.575 

7.795 

1.800 

8.769 

2.025 

9.744 

2.250 

77    0 

15 

5.840 

1.375 

6.814 

1.604 

7.787 

1.834 

8.760 

2.063 

9.734 

2.292 

45 

30 

5.834 

1.401 

6.807 

1.634 

7.779 

1.868 

8.751 

2.101 

9.724 

2.335 

30 

45 

5.828 

1.426 

6.799 

1.664 

7.771 

1.902 

8.742 

2.139 

9.713 

2.377 

15 

14    0 

5.822 

1.452 

6.792 

1.693 

7.762 

1.935 

8.733 

2.177 

9.703 

2.419 

76   0 

15 

5.815 

1.477 

6.785 

1.723 

7.754 

1.969 

8.723 

2.215 

9.692 

2.462 

45 

30 

5.809 

1.502 

6.777 

1.753 

7.745 

2.003 

8.713 

2.253 

9.682 

2.504 

30 

45 

5.802 

1.528 

6.769 

1.782 

7.736 

2.037 

8.703 

2.291 

9.671 

2.546 

15 

15    0 

5.796 

1.553 

6.761 

1.812 

7.727 

2.071 

8.693 

2.329 

9.659 

2.588 

75   0 

o      f 

Dep.     Lat. 
Distance  6. 

Dep.       Lat. 
Distance  7. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o      f 

Bearing. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

76° -90 


72 

16°- 

-30° 

Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

o      f 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

O        f 

15  15 

0.965 

0.263 

1.930 

0.526 

2.894 

0.789 

3.859 

1.052 

4.824 

1.315 

74  45 

30 

0.964 

0.267 

1.927 

0.534 

2.891 

0.802 

3.855 

1.069 

4.818 

1.336 

30 

45 

0.962 

0.271 

1.925 

0.543 

2.887 

0.814 

3.850 

1.086 

4.812 

1.357 

15 

16   0 

0.961 

0.276 

1.923 

0.551 

2.884 

0.827 

3.845 

1.103 

4.806 

1.378 

74   0 

15 

0.960 

0.280 

1.920 

0.560 

2.880 

0.839 

3.840 

1.119 

4.800 

1.399 

45 

30 

0.959 

0.284 

1.918 

0.568 

2.876 

0.852 

3.835 

1.136 

4.794 

1.420 

30 

45 

0.958 

0.288 

1.915 

0.576 

2.873 

0.865 

3.830 

1.153 

4.788 

1.441 

15 

17    0 

0.956 

0.292 

1.913 

0.585 

2.869 

0.877 

3.825 

1.169 

4.782 

1.462 

73    0 

15 

0.955 

0.297 

1.910 

0.593 

2.865 

0.890 

3.820 

1.186 

4.775 

1.483 

45 

30 

0.954 

0.301 

1.907 

0.601 

2.861 

0.902 

3.815 

1.203 

4.769 

1.504 

30 

45 

0.952 

0.305 

1.905 

0.610 

2.857 

0.915 

3.810 

1.220 

4.762 

1.524 

15 

18    0 

0.951 

0.309 

1.902 

0.618 

2.853 

0.927 

3.804 

1.236 

4.755 

1.545 

72    0 

15 

0.950 

0.313 

1.899 

0.626 

2.849 

0.939 

3.799 

1.253 

4.748 

1.566 

45 

30 

0.948 

0.317 

1.897 

0.635 

2.845 

0.952 

3.793 

1.269 

4.742 

1.587 

30 

45 

0.947 

0.321 

1.894 

0.643 

2.841 

0.964 

3.788 

1.286 

4.735 

1.607 

15 

19    0 

0.946 

0.326 

1.891 

0.651 

2.837 

0.977 

3.782 

1.302 

4.728 

1.628 

71    0 

15 

0.944 

0.330 

1.888 

0.659 

2.832 

0.989 

3.776 

1.319 

4.720 

1.648 

45 

30 

0.943 

0.334 

1.885 

0.668 

2.828 

1.001 

3.771 

1.335 

4.713 

1.669 

30 

45 

0.941 

0.338 

1.882 

0.676 

2.824 

1.014 

3.765 

1.352 

4.706 

1.690 

15 

20   0 

0.940 

0.342 

1.879 

0.684 

2.819 

1.026 

3.759 

1.368 

4.698 

1.710 

70   0 

15 

0.938 

0.346 

1.876 

0.692 

2.815 

1.038 

3.753 

1.384 

4.691 

1.731 

45 

30 

0.937 

0.350 

1.873 

0.700 

2.810 

1.051 

3.747 

1.401 

4.683 

1.751 

30 

45 

0.935 

0.354 

1.870 

0.709 

2.805 

1.063 

3.741 

1.417 

4.676 

1.771 

15 

21    0 

0.934 

0.358 

1.867 

0.717 

2.801 

1.075 

3.734 

1.433 

4.668 

1.792 

69    0 

15 

0.932 

0.362 

1.864 

0.725 

2.796 

1.087 

3.728 

1.450 

4.660 

1.812 

45 

30 

0.930 

0.367 

1.861 

0.733 

2.791 

1.100 

3.722 

1.466 

4.652 

1.833 

30 

45 

0.929 

0.371 

1.858 

0.741 

2.786 

1.112 

3.715 

1.482 

4.644 

1.853 

15 

22    0 

0.927 

0.375 

1.854 

0.749 

2.782 

1.124 

3.709 

1.498 

4.636 

1.873 

68   0 

15 

0.926 

0.379 

1.851 

0.757 

2.777 

1.136 

3.702 

1.515 

4.628 

1.893 

45 

30 

0.924 

0.383 

1.848 

0.765 

2.772 

1.148 

3.696 

1.531 

4.619 

1.913 

30 

45 

0.922 

0.387 

1.844 

0.773 

2.767 

1.160 

3.689 

1.547 

4.611 

1.934 

15 

23   0 

0.921 

0.391 

1.841 

0.781 

2.762 

1.172 

3.682 

1.563 

4.603 

1.954 

67    0 

15 

0.919 

0.395 

1.838 

0.789 

2.756 

1.184 

3.675 

1.579 

4.594 

1.974 

45 

30 

0.917 

0.399 

1.834 

0.797 

2.751 

1.196 

3.668 

1.595 

4.585 

1.994 

30 

45 

0.915 

0.403 

1.831 

0.805 

2.746 

1.208 

3.661 

1.611 

4.577 

2.014 

15 

24   0 

0.914 

0.407 

1.827 

0.813 

2.741 

1.220 

3.654 

1.627 

4.568 

2.034 

66   0 

15 

0.912 

0.411 

1.824 

0.821 

2.735 

1.232 

3.647 

1.643 

4.559 

2.054 

45 

30 

0.910 

0.415 

1.820 

0.829 

2.730 

1.244 

3.640 

1.659 

4.550 

2.073 

30 

45 

0.908 

0.419 

1.816 

0.837 

2.724 

1.256 

3.633 

1.675 

4.541 

2.093 

15 

25   0 

0.906 

0.423 

1.813 

0.845 

2.719 

1.268 

3.625 

1.690 

4.532 

2.113 

65   0 

15 

0.904 

0.427 

1.809 

0.853 

2.713 

1.280 

3.618 

1.706 

4.522 

2.133 

45 

30 

0.903 

0.431 

1.805 

0.861 

2.708 

1.292 

3.610 

1.722 

4.513 

2.153 

30 

45 

0.901 

0.434 

1.801 

0.869 

2.702 

1.303 

3.603 

1.738 

4.503 

2.172 

15 

26   0 

0.899 

0.438 

1.798 

0.877 

2.696 

1.315 

3.595 

1.753 

4.494 

2.192 

64    0 

15 

0.897 

0.442 

1.794 

0.885 

2.691 

1.327 

3.587 

1.769 

4.484 

2.211 

45 

30 

0.895 

0.446 

1.790 

0892 

2.685 

1.339 

3.580 

1.785 

4.475 

2.231 

30 

45 

0.893 

0.450 

1.786 

0.900 

2.679 

1.350 

3.572 

1.800 

4.465 

2.250 

15 

27    0 

0.891 

0.454 

1.782 

0.908 

2.673 

1.362 

3.564 

1.816 

4.455 

2.270 

63    0 

15 

0.889 

0.458 

1.778 

0.916 

2.667 

1.374 

3.556 

1.831 

4.445 

2.289 

45 

30 

0.887 

0.462 

1.774 

0.923 

2.661 

1.385 

3.548 

1.847 

4.435 

2.309 

30 

45 

0.885 

0.466 

1.770 

0.931 

2.655 

1.397 

3.540 

1.862 

4.425 

2.328 

15 

28    0 

0.883 

0.469 

1.766 

0.939 

2.649 

1.408 

3.532 

1.878 

4.415 

2.347 

62    0 

15 

0.881 

0.473 

1.762 

0.947 

2.643 

1.420 

3.524 

1.893 

4.404 

2.367 

45 

30 

0.879 

0.477 

1.758 

0.954 

2.636 

1.431 

3.515 

1.909 

4.394 

2.386 

30 

45 

0.877 

0.481 

1.753 

0.962 

2.630 

1.443 

3.507 

1.924 

4.384 

2.405 

15 

29   0 

0.875 

0.485 

1.749 

0.970 

2.624 

1.454 

3.498 

1.939 

4.373 

2.424 

61    0 

15 

0.872 

0.489 

1.745 

0.977 

2.617 

1.466 

3.490 

1.954 

4.362 

2.443 

45 

30 

0.870 

0.492 

1.741 

0.985 

2.611 

1.477 

3.481 

1.970 

4.352 

2.462 

30 

45 

0.868 

0.496 

1.736 

0.992 

2.605 

1.489 

3.473 

1.985 

4.341 

2.481 

15 

30   0 

0.866 

0.500 

1.732 

1.000 

2.598 

1.500 

3.464 

2.000 

4.330 

2.500 

60   0 

o    r 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

O        f 

Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

60° -76' 


16°- 

-30 

3 

73 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

o    r 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.      Dep. 

Lat. 

Dep. 

o     r 

15  15 

5.789 

1.578 

6.754 

1.841 

7.718 

2.104 

8.683   2.367 

9.648 

2.630 

74  45 

30 

5.782 

1.603 

6.745 

1.871 

7.709 

2.138 

8.673   2.405 

9.636 

2.672 

30 

45 

5.775 

1.629 

6.737 

1.900 

7.700 

2.172 

8.662   2.443 

9.625 

2.714 

15 

16    0 

5.768 

1.654 

6.729 

1.929 

7.690 

2.205 

8.651   2.481 

9.613 

2.756 

74    0 

15 

5.760 

1.679 

6.720 

1.959 

7.680 

2.239 

8.640  2.518 

9.601 

2.798 

45 

30 

5.753 

1.704 

6.712 

1.988 

7.671 

2.272 

8.629  2.556 

9.588 

2.840 

30 

45 

5.745 

1.729 

6.703 

2.017 

7.661 

2.306 

8.618   2.594 

9.576 

2.882 

15 

17    0 

5.738 

1.754 

6.694 

2.047 

7.650 

2.339 

8.607   2.631 

9.563 

2.924 

73    0 

15 

5.730 

1.779 

6.685 

2.076 

7.640 

2.372 

8.595   2.669 

9.550 

2.965 

45 

30 

5.722 

1.804 

6.676 

2.105 

7.630 

2.406 

8.583   2.706 

9.537 

3.007 

30 

45 

5.714 

1.829 

6.667 

2.134 

7.619 

2.439 

8.572   2.744 

9.524 

3.049 

15 

18    0 

5.706 

1.854 

6.657 

2.163 

7.608 

2.472 

8.560  2.781 

9.511 

3.090 

72    0 

15 

5.698 

1.879 

6.648 

2.192 

7.598 

2.505 

8.547   2.818 

9.497 

3.132 

45 

30 

5.690 

1.904 

6.638 

2.221 

7.587 

2.538 

8.535   2.856 

9.483 

3.173 

30 

45 

5.682 

1.929 

6.629 

2.250 

7.575 

2.572 

8.522   2.893 

9.469 

3.214 

15 

19    0 

5.673 

1.953 

6.619 

2.279 

7.564 

2.605 

8.510  2.930 

9.455 

3.256 

71    0 

15 

5.665 

1.978 

6.609 

2.308 

7.553 

2.638 

8.497   2.967 

9.441 

3.297 

45 

30 

5.656 

2.003 

6.598 

2.337 

7.541 

2.670 

8.484  3.004 

9.426 

3.338 

30 

45 

5.647 

2.028 

6.588 

2.365 

7.529 

2.703 

8.471   3.041 

9.412 

3.379 

15 

20   0 

5.638 

2.052 

6.578 

2.394 

7.518 

2.736 

8.457  3.078 

9.397 

3.420 

70   0 

15 

5.629 

2.077 

6.567 

2.423 

7.506 

2.769 

8.444  3.115 

9.382 

3.461 

45 

30 

5.620 

2.101 

6.557 

2.451 

7.493 

2.802 

8.430  3.152 

9.367 

3.502 

30 

45 

5.611 

2.126 

6.546 

2.480 

7.481 

2.834 

8.416  3.189 

9.351 

3.543 

15 

21    0 

5.601 

2.150 

6.535 

2.509 

7.469 

2.867 

8.402  3.225 

9.336 

3.584 

69   0 

15 

5.592 

2.175 

6.524 

2.537 

7.456 

2.900 

8.388  3.262 

9.320 

3.624 

45 

30 

5.582 

2.199 

6.513 

2.566 

7.443 

2.932 

8.374  3.299 

9.304 

3.665 

30 

45 

5.573 

2.223 

6.502 

2.594 

7.430 

2.964 

8.359  3.335 

9.288 

3.706 

15 

22    0 

5.563 

2.248 

6.490 

2.622 

7.417 

2.997 

8.345  3.371 

9.272 

3.746 

68   0 

15 

5.553 

2.272 

6.479 

2.651 

7.404 

3.029 

8.330  3.408 

9.255 

3.787 

45 

30 

5.543 

2.296 

6.467 

2.679 

7.391 

3.061 

8.315   3.444 

9.239 

3.827 

30 

45 

5.533 

2.320 

6.455 

2.707 

7.378 

3.094 

8.300  3.480 

9.222 

3.867 

15  i 

23   0 

5.523 

2.344 

6.444 

2.735 

7.364 

3.126 

8.285   3.517 

9.205 

3.907 

67    0 

15 

5.513 

2.368 

6.432 

2.763 

7.350 

3.158 

8.269  3.553 

9.188 

3.947 

45 

30 

5.502 

2.392 

6.419 

2.791 

7.336 

3.190 

8.254  3.589 

9.171 

3.988 

30 

45 

5.492 

2.416 

6.407 

2.819 

7.322 

3.222 

8.238  3.625 

9.153 

4.028 

15 

24   0 

5.481 

2.440 

6.395 

2.847 

7.308 

3.254 

8.222  3.661 

9.136 

4.067 

66   0 

15 

5.471 

2.464 

6.382 

2.875 

7.294 

3.286 

8.206  3.696 

9.118 

4.107 

45 

30 

5.460 

2.488 

6.370 

2.903 

7.280 

3.318 

8.190  3.732 

9.100 

4.147 

30 

45 

5.449 

2.512 

6.357 

2.931 

7.265 

3.349 

8.173  3.768 

9.081 

4.187 

15 

25   0 

5.438 

2.536 

6.344 

2.958 

7.250 

3.381 

8.157  3.804 

9.063 

4.226 

65    0 

15 

5.427 

2.559 

6.331 

2.986 

7.236 

3.413 

8.140  3.839 

9.045 

4.266 

45 

30 

5.416 

2.583 

6.318 

3.014 

7.221 

3.444 

8.123   3.875 

9.026 

4.305 

30 

45 

5.404 

2.607 

6.305 

3.041 

7.206 

3.476 

8.106  3.910 

9.007 

4.345 

15 

26   0 

5.393 

2.630 

6.292 

3.069 

7.190 

3.507 

8.089  3.945 

8.988 

4.384 

64   0 

15 

5.381 

2.654 

6.278 

3.096 

^7.175 

3.538 

8.072   3.981 

8.969 

4.423 

45 

30 

5.370 

2.677 

6.265 

3.123 

7.160 

3.570 

8.054  4.016 

8.949 

4.462 

30 

45 

5.358 

2.701 

6.251 

3.151 

7.144 

3.601 

8.037   4.051 

8.930 

4.501 

15 

27    0 

5.346 

2.724 

6.237 

3.178 

7.128 

3.632 

8.019  4.086 

8.910 

4.540 

63    0 

15 

5.334 

2.747 

6.223 

3.205 

7.112 

3.663 

8.001   4.121 

8.890 

4.579 

45 

30 

5.322 

2.770 

6.209 

3.232 

7.096 

3.694 

7.983  4.156 

8.870 

4.618 

30 

45 

5.310 

2.794 

6.195 

3.259 

7.080 

3.725 

7.965   4.190 

8.850 

4.656 

15 

28   0 

5.298 

2.817 

6.181 

3.286 

7.064 

3.756 

7.947  4.225 

8.829 

4.695 

62    0 

15 

5.285 

2.840 

6.166 

3.313 

7.047 

3.787 

7.928  4.260 

8.809 

4.733 

45 

30 

5.273 

2.863 

6.152 

3.340 

7.031 

3.817 

7.909  4.294 

8.788 

4.772 

30 

45 

5.260 

2.886 

6.137 

3.367 

7.014 

3.848 

7.891   4.329 

8.767 

4.810 

15 

29   0 

5.248 

2.909 

6.122 

3.394 

6.997 

3.878 

7.872  4.363 

8.746 

4.848 

61    0 

15 

5.235 

2.932 

6.107 

3.420 

6.980 

3.909 

7.852  4.398 

8.725 

4.886 

45 

30 

5.222 

2.955 

6.093 

3.447 

6.963 

3.939 

7.833   4.432 

8.704 

4.924 

30 

45 

5.209 

2.977 

6.077 

3.474 

6.946 

3.970 

7.814  4.466 

8.682 

4.962 

15 

30    0 

5.196 

3.000 

6.062 

3.500 

6  928 

4.000 

7.794  4.500 

8.660 

5.000 

60   0 

o      f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.      Lat. 

Dep. 

Lat. 

o      f 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

60° -76^ 


T4 

30°- 

-46 

O 

Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing, 

O       f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o     t 

30  15 

0.864 

0.504 

1.728 

1.008 

2.592 

1.511 

3.455 

2.015 

4.319 

2.519 

59  45 

30 

0.862 

0.508 

1.723 

1.015 

2.585 

1.523 

3.447 

2.030 

4.308 

2.538 

30 

45 

0.859 

0.511 

1.719 

1.023 

2.578 

1.534 

3.438 

2.045 

4.297 

2.556 

15 

31    0 

0.857 

0.515 

1.714 

1.030 

2.572 

1.545 

3.429 

2.060 

4.286 

2.575 

59   0 

15 

0.855 

0.519 

1.710 

1.038 

2.565 

1.556 

3.420 

2.075 

4.275 

2.594 

45 

30 

0.853 

0.522 

1.705 

1.045 

2.558 

1.567 

3.411 

2.090 

4.263 

2.612 

30 

45 

0.850 

0.526 

1.701 

1.052 

2.551 

1.579 

3.401 

2.105 

4.252 

2.631 

15 

32    0 

0.848 

0.530 

1.696 

1.060 

2.544 

1.590 

3.392 

2.120 

4.240 

2.650 

58   0 

15 

0.846 

0.534 

1.691 

1.067 

2.537 

1.601 

3.383 

2.134 

4.229 

2.668 

45 

30 

0.843 

0.537 

1.687 

1.075 

2.530 

1.612 

3.374 

2.149 

4.217 

2.686 

30 

45 

0.841 

0.541 

1.682 

1.082 

2.523 

1.623 

3.364 

2.164 

4.205 

2.705 

15 

33    0 

0.839 

0.545 

1.677 

1.089 

2.516 

1.634 

3.355 

2.179 

4.193 

2.723 

57    0 

15 

0.836 

0.548 

1.673 

1.097 

2.509 

1.645 

3.345 

2.193 

4.181 

2.741 

45 

30 

0.834 

0.552 

1.668 

1.104 

2.502 

1.656 

3.336 

2.208 

4.169 

2.760 

30 

45 

0.831 

0.556 

1.663 

1.111 

2.494 

1.667 

3.326 

2.222 

4.157 

2.778 

15 

34   0 

0.829 

0.559 

1.658 

1.118 

2.487 

1.678 

3.316 

2.237 

4.145 

2.796 

56   0 

15 

0.827 

0.563 

1.653 

1.126 

2.480 

1.688 

3.306 

2.251 

4.133 

2.814 

45 

30 

0.824 

0.566 

1.648 

1.133 

2.472 

1.699 

3.297 

2.266 

4.121 

2.832 

30 

45 

0.822 

0.570 

1.643 

1.140 

2.465 

1.710 

3.287 

2.280 

4.108 

2.850 

15 

35   0 

0.819 

0.574 

1.638 

1.147 

2.457 

1.721 

3.277 

2.294 

4.096 

2.868 

^^    0 

15 

0.817 

0.577 

1.633 

1.154 

2.450 

1.731 

3.267 

2.309 

4.083 

2.886 

45 

30 

0.814 

0.581 

1.628 

1.161 

2.442 

1.742 

3.257 

2.323 

4.071 

2.904 

30 

45 

0.812 

0.584 

1.623 

1.168 

2.435 

1.753 

3.246 

2.337 

4.058 

2.921 

15 

36   0 

0.809 

0.588 

1.618 

1.176 

2.427 

1.763 

3.236 

2.351 

4.045 

2.939 

54   0 

15 

0.806 

0.591 

1.613 

1.183 

2.419 

1.774 

3.226 

2.365 

4.032 

2.957 

45 

30 

0.804 

0.595 

1.608 

1.190 

2.412 

1.784 

3.215 

2.379 

4.019 

2.974 

30 

45 

0.801 

0.598 

1.603 

1.197 

2.404 

1.795 

3.205 

2.393 

4.006 

2.992 

15 

37    0 

0.799 

0.602 

1.597 

1.204 

2.396 

1.805 

3.195 

2.407 

3.993 

3.009 

53   0 

15 

0.796 

0.605 

1.592 

1.211 

2.388 

1.816 

3.184 

2.421 

3.980 

3.026 

45 

30 

0.793 

0.609 

1.587 

1.218 

2.380 

1.826 

3.173 

2.435 

3.967 

3.044 

30 

45 

0.791 

0.612 

1.581 

1.224 

2.372 

1.837 

3.163 

2.449 

3.953 

3.061 

15 

38   0 

0.788 

0.616 

1.576 

1.231 

2.364 

1.847 

3.152 

2.463 

3.940 

3.078 

52   0 

15 

0.785 

0.619 

1.571 

1.238 

2.356 

1.857 

3.141 

2.476 

3.927 

3.095 

45 

30 

0.783 

0.623 

1.565 

1.245 

2.348 

1.868 

3.130 

2.490 

3.913 

3.113 

30 

45 

0.780 

0.626 

1.560 

1.252 

2.340 

1.878 

3.120 

2.504 

3.899 

3.130 

15 

39   0 

0.777 

0.629 

1.554 

1.259 

2.331 

1.888 

3.109 

2.517 

3.886 

3.147 

51    0 

15 

0.774 

0.633 

1.549 

1.265 

2.323 

1.898 

3.098 

2.531 

3.872 

3.164 

45 

30 

0.772 

0.636 

1.543 

1.272 

2.315 

1.908 

3.086 

2.544 

3.858 

3.180 

30 

45 

0.769 

0.639 

1.538 

1.279 

2.307 

1.918 

3.075 

2.558 

3.844 

3.197 

15 

40   0 

0.766 

0.643 

1.532 

1.286 

2.298 

1.928 

3.064 

2.571 

3.830 

3.214 

50   0 

15 

0.763 

0.646 

1.526 

1.292 

2.290 

1.938 

3.053 

2.584 

3.816 

3.231 

45 

30 

0.760 

0.649 

1.521 

1.299 

2.281 

1.948 

3.042 

2.598 

3.802 

3.247 

30 

45 

0.758 

0.653 

1.515 

1.306 

2.273 

1.958 

3.030 

2.611 

3.788 

3.264 

15 

41    0 

0.755 

0.656 

1.509 

1.312 

2.264 

1.968 

3.019 

2.624 

3.774 

3.280 

49   0 

15 

0.752 

0.659 

1.504 

1.319 

2.256 

1.978 

3.007 

2.637 

3.759 

3.297 

45 

30 

0.749 

0.663 

1.498 

1.325 

2.247 

1.988 

2.996 

2.650 

3.745 

3.313 

30 

45 

0.746 

0.666 

1.492 

1.332 

2.238 

1.998 

2.984 

2.664 

3.730 

3.329 

15 

42    0 

0.743 

0.669 

1.486 

1.338 

2.229 

2.007 

2.973 

2.677 

3.716 

3.346 

48    0 

15 

0.740 

0.672 

1.480 

1.345 

2.221 

2.017 

2.961 

2.689 

3.701 

3.362 

45 

30 

0.737 

0.676 

1.475 

1.351 

2.212 

2.027 

2.949 

2.702 

3.686 

3.378 

30 

45 

0.734 

0.679 

1.469 

1.358 

2.203 

2.036 

2.937 

2.715 

3.672 

3.394 

15 

43   0 

0.731 

0.682 

1.463 

1.364 

2.194 

2.046 

2.925 

2.728 

3.657 

3.410 

47    0 

15 

0.728 

0.685 

1.457 

1.370 

2.185 

2.056 

2.913 

2.741 

3.642 

3.426 

45 

30 

0.725 

0.688 

1.451 

1.377 

2.176 

2.065 

2.901 

2.753 

3.627 

3.442 

30 

45 

0.722 

0.692 

1.445 

1.383 

2.167 

2.075 

2.889 

2.766 

3.612 

3.458 

15 

44    0 

0.719 

0.695 

1.439 

1.389 

2.158 

2.084 

2.877 

2.779 

3.597 

3.473 

46   0 

15 

0.716 

0.698 

1.433 

1.396 

2.149 

2.093 

2.865 

2.791 

3.582 

3.489 

45 

30 

0.713 

0.701 

1.427 

1.402 

2.140 

2.103 

2.853 

2.804 

3.566 

3.505 

30 

45 

0.710 

0.704 

1.420 

1.408 

2.131 

2.112 

2.841 

2.816 

3.551 

3.520 

15 

45    0 

0.707 

0.707 

1.414 

1.414 

2.121 

2.121 

2.828 

2.828 

3.536 

3.536 

45    0 

O       f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o      f 

Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

46° -60^ 


30°- 

-46 

o 

76 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

o      f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.      Dep. 

Lat.      Dep. 

o     t 

3015 

5.183 

3.023 

6.047 

3.526 

6.911 

4.030 

7.775  4.534 

8.638  5.038 

59  45 

30 

5.170 

3.045 

6.031 

3.553 

6.893 

4.060 

7.755  4.568 

8.616  5.075 

30 

45 

5.156 

3.068 

6.016 

3.579 

6.875 

4.090 

7.735  4.602 

8.594   5.113 

15 

31    0 

5.143 

3.090 

6.000 

3.605 

6.857 

4.120 

7.715  4.635 

8.572   5.150 

59   0 

15 

5.129 

3.113 

5.984 

3.631 

6.839 

4.150 

7.694  4.669 

8.549  5.188 

45 

30 

5.116 

3.135 

5.968 

3.657 

6.821 

4.180 

7.674   4.702 

8.526  5.225 

30 

45 

5.102 

3.157 

5.952 

3.683 

6.803 

4.210 

7.653   4.736 

8.504   5.262 

15 

32    0 

5.088 

3.180 

5.936 

3.709 

6.784 

4.239 

7.632  4.769 

8.481   5.299 

58   0 

15 

5.074 

3.202 

5.920 

3.735 

6.766 

4.269 

7.612  4.802 

8.457  5.336 

45 

30 

5.060 

3.224 

5.904 

3.761 

6.747 

4.298 

7.591   4.836 

8.434  5.373 

30 

45 

5.046 

3.246 

5.887 

3.787 

6.728 

4.328 

7.569  4.869 

8.410  5.410 

15 

33   0 

5.032 

3.268 

5.871 

3.812 

6.709 

4.357 

7.548  4.902 

8.387   5.446 

57   0 

15 

5.018 

3.290 

5.854 

3.838 

6.690 

4.386 

7.527  4.935 

8.363  5.483 

45 

30 

5.003 

3.312 

5.837 

3.864 

6.671 

4.416 

7.505   4.967 

8.339  5.519 

30 

45 

4.989 

Z3ZZ 

5.820 

3.889 

6.652 

4.445 

7.483  5.000 

8.315  5.556 

15 

34   0 

4.974 

3.355 

5.803 

3.914 

6.632 

4.474 

7.461   5.033 

8.290  5.592^ 

56   0 

15 

4.960 

3.377 

5.786 

3.940 

6.613 

4.502 

7.439  5.065 

8.266  5.628 

45 

30 

4.945 

3.398 

5.769 

3.965 

6.593 

4.531 

7.417  5.098 

8.241   5.664 

30 

45 

4.930 

3.420 

5.752 

3.990 

6.573 

4.560 

7.395   5.130 

8.217  5.700 

15 

35   0 

4.915 

3.441 

5.734 

4.015 

6.553 

4.589 

7.372  5.162 

8.192  5.736 

55   0 

15 

4.900 

3.463 

5.716 

4.040 

6.533 

4.617 

7.350  5.194 

8.166  5.772 

45 

30 

4.885 

3.484 

5.699 

4.065 

6.513 

4.646 

7.327  5.226 

8.141  5.807 

30 

45 

4.869 

3.505 

5.681 

4.090 

6.493 

4.674 

7.304  5.258 

8.116  5.843 

15 

36   0 

4.854 

3.527 

5.663 

4.115 

6.472 

4.702 

7.281   5.290 

8.090  5.878 

54   0 

15 

4.839 

3.548 

5.645 

4.139 

6.452 

4.730 

7.258   5.322 

8.064  5.913 

45 

30 

4.823 

3.569 

5.627 

4.164 

6.431 

4.759 

7.235   5.353 

8.039  5.948 

30 

45 

4.808 

3.590 

5.609 

4.188 

6.410 

4.787 

7.211   5.385 

8.013  5.983 

15 

37    0 

4.792 

3.611 

5.590 

4.213 

6.389 

4.815 

7.188  5.416 

7.986  6.018 

53   0 

15 

4.776 

3.632 

5.572 

4.237 

6.368 

4.842 

7.164  5.448 

7.960  6.053 

45 

30 

4.760 

3.653 

5.554 

4.261 

6.347 

4.870 

7.140  5.479 

7.934  6.088 

30 

45 

4.744 

3.673 

5.535 

4.286 

6.326 

4.898 

7.116  5.510 

7.907  6.122 

15 

38   0 

4.728 

3.694 

5.516 

4.310 

6.304 

4.925 

7.092  5.541 

7.880  6.157 

52   0 

15 

4.712 

3.715 

5.497 

4.334 

6.283 

4.953 

7.068  5.572 

7.853  6.191 

45 

30 

4.696 

3.735 

5.478 

4.358 

6.261 

4.980 

7.043   5.603 

7.826  6.225 

30 

45 

4.679 

3.756 

5.459 

4.381 

6.239 

5.007 

7.019  5.633 

7.799  6.259 

15 

39   0 

4.663 

3.776 

5.440 

4.405 

6.217 

5.035 

6.994  5.664 

7.772  6.293 

51    0 

15 

4.646 

3.796 

5.421 

4.429 

6.195 

5.062 

6.970  5.694 

7.744  6.327 

45 

30 

4.630 

3.816 

5.401 

4.453 

6.173 

5.089 

6.945   5.725 

7.716  6.361 

30 

45 

4.613 

3.837 

5.382 

4.476 

6.151 

5.116 

6.920  5.755 

7.688  6.394 

15 

40   0 

4.596 

3.857 

5.362 

4.500 

6.128 

5.142 

6.894  5.785 

7.660  6.428 

50   0 

15 

4.579 

3.877 

5.343 

4.523 

6.106 

5.169 

6.869  5.815 

7.632  6.461 

45 

30 

4.562 

3.897 

5.323 

4.546 

6.083 

5.196 

6.844  5.845 

7.604  6.495 

.    30 

45 

4.545 

3.917 

5.303 

4.569 

6.061 

5.222 

6.818   5.875 

7.576  6.528 

15 

41    0 

4.528 

3.936 

5.283 

4.592 

6.038 

5.248 

6.792   5.905 

7.547  6.561 

49    0 

15 

4.511 

3.956 

5.263 

4.615 

6.015 

5.275 

6.767   5.934 

7.518  6.594 

45 

30 

4.494 

3.976 

5.243 

4  638 

5.992 

5.301 

6.741   5.964 

7.490  6.626 

30 

45 

4.476 

3.995 

5.222 

4.661 

5.968 

5.327 

6.715   5.993 

7.461   6.659 

•  15 

42    0 

4.459 

4.015 

5.202 

4.684 

5.945 

5.353 

6.688  6.022 

7.431   6.691 

48    0 

15 

4.441 

4.034 

5.182 

4.707 

5.922 

5.379 

6.662  6.051 

7.402  6.724 

45 

30 

4.424 

4.054 

5.161 

4.729 

5.898 

5.405 

6.635  6.080 

7.373  6.756 

30 

45 

4.406 

4.073 

5.140 

4.752 

5.875 

5.430 

6.609  6.109 

7.343  6.788 

15 

43   0 

4.388 

4.092 

5.119 

4.774 

5.851 

5.456 

6.582  6.138 

7.314  6.820 

47   0 

15 

4.370 

4.111 

5.099 

4.796 

5.827 

5.481 

6.555  6.167 

7.284  6.852 

45 

30 

4.352 

4.130 

5.078 

4.818 

5.803 

5.507 

6.528  6.195 

7.254  6.884 

30 

45 

4.334 

4.149 

5.057 

4.841 

5.779 

5.532 

6.501   6.224 

7.224  6.915 

15 

44   0 

4.316 

4.168 

5.035 

4.863 

5.755 

5.557 

6.474  6.252 

7.193  6.947 

46   0 

15 

4.298 

4.187 

5.014 

4.885 

5.730 

5.582 

6.447  6.280 

7.163  6.978 

45 

30 

4.280 

4.206 

4.993 

4.906 

5.706 

5.607 

6.419  6.308 

7.133   7.009 

30 

45 

4.261 

4.224 

4.971 

4.928 

5.681 

5.632 

6.392  6.336 

7.102  7.040 

15 

45    0 

4.243 

4.243 

4.950 

4.950 

5.657 

5.657 

6.364  6.364 

7.071   7.071 

45   0 

o    r 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.     Lat. 

Dep.     Lat. 

Q    ' 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9.  Distance  10. 

Bearing. 

46° -60' 


A  TABLE   OF  THE   ANGLES 

Which  every  Point  and  Quarter  Point  of  the  Compass  makes  with  the  Meridian. 


North. 

Points. 

0        ^       // 

2  48  45 

5  37  30 

8  26  15 

11  15    0 

Points. 

0-14 
0-% 
0-34 

1 

South.                  1 

N.  by  E. 

N.  by  W. 

S.  by  E. 

S.  by  -W. 

N.N.E. 

N.N.W. 

I-V4 

14    3  45 
16  52  30 
19  41  15 
22  30    0 

2      * 

S.S.E. 

S.S.W. 

N.E,  by  N. 

N.-W.  by  N. 

25  18  45 
28     7  30 
30  56  15 
33  45    0 

3      * 

S.E.byS. 

S.W.  by  S. 

N.E. 

N.^W. 

3-V4 

36  33  45 
39  22  30 
42  11  15 
45    0    0 

3-1/4 
3-% 

.A 

S.E. 

S.'W. 

N.E.  by  E 

N.W.by-W. 

P 

47  48  45 
50  37  30 
53  26  15 
56  15    0 

4-14 
4-% 
4-3/ 
5 

S.E.byE. 

S.W.  by  "W. 

E.N.E. 

W.N.W. 

P 

59    3  45 
61  52  30 
64  41  15 
67  30    0 

5-1/4 
6      * 

E.S.E. 

W.S.W. 

E.  by  N. 

^W.  by  N. 

6-y. 

7    * 

70  18  45 
73    7  30 
75  56  15 
78  45    0 

6 -1/4 
7 

E.  by  S. 

W.  by  S. 

East. 

"West. 

7-V4 

81  33  Ah 
84  22  30 
87  11  15 
90    0    0 

7-V4 

East. 

West. 

12 


Oc»' 


'•^ 


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